Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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127 
128 \title{\textbf{Projector Resolved Seed Calculus IV}\\[0.45em]
129 \large Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data}
130 \author{Komorebi P.}
131 \date{Modern Monograph, 8 October 2026}
132 
133 \hypersetup{
134   pdftitle={Projector-Resolved Seed Calculus IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data},
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136   pdfsubject={Cyclic exponential sections, reciprocal families, zero geometry, and spectral zero data},
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170 \begin{document}
171 \KomorebiCover
172 \KomorebiOpeningNote{Begin with cyclic projection and the signed differential cycle. The reciprocal chapters then build parameter polynomials, Appell systems, and theta polynomials from a unit series. The second half follows the same denominator through large zero geometry, canonical products, and zero zeta functions.
173 
174 The visual atlas at the end of this volume supplies twenty-six worked plates. The separate Lean guide follows the released declarations and their hypotheses. All sixty-eight proof bodies and ninety-five numbered records of the selected manuscript are retained.}
175 
176 \begin{abstract}
177 Fix a positive integer $N$. Root of unity projection of the exponential
178 produces residue-class sections that, for $\lambda\ne0$, form a basis for
179 $y^{(N)}=\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the
180 denominator $\Dfun_N$ and its reduced form $\Delta_N$. We derive reciprocal
181 coefficient polynomials, Bell and Newton recurrences, an additive parameter
182 law, and residue section Appell and theta families. For $N\geq2$,
183 $\Delta_N$ has order $1/N$. We specialize classical two term zero theory for
184 exponential sums to the regular exponent polygon. In this equal-weight family,
185 conjugation symmetry combined with unique near-model localization places all
186 sufficiently large zeros exactly on the symmetry rays, and direct expansion
187 gives their leading displacement and derivative sign. The classical perimeter
188 law yields zero-counting asymptotics with explicit leading constants and
189 bounded error. These results give genus zero products, the standard diagonal
190 Fredholm representation, zero-power-sum formulas for reciprocal and
191 logarithmic coefficients, nearest-zero coefficient asymptotics, and
192 meromorphic continuation of the symmetry-orbit zero zeta by comparison with
193 the Hurwitz zeta function. Complex powers are interpreted either formally or
194 through a fixed logarithm on a zero free domain. The paper treats only this
195 cyclic analytic family. Arithmetic interpolation, general nonlinear trees,
196 general resolvent theory, and abstract completion are reserved for Papers IX,
197 III, VI, and X.
198 \end{abstract}
199 
200 \begin{readerroadmap}[title={Formal scope and evidence}]
201 The preserved Paper IV v1.0.0 release records full scoped Lean coverage of the sixty-eight theorem like manuscript results under their original hypotheses. This volume changes presentation and adds worked explanations. It does not create a new formal release.
202 
203 Imported standard fact 11.1 is used in the ordinary manuscript growth argument. The Lean growth-order proofs establish that step separately. Open questions, branch choices, multiplicities, and the hypotheses of coefficient asymptotics remain explicit. Current finite computations support the worked models. The compiler and kernel evidence remains the preserved September release evidence.
204 \end{readerroadmap}
205 
206 \clearpage
207 \tableofcontents
208 \listoffigures
209 \clearpage\pagenumbering{arabic}
210 
211 \section{Introduction}\label{sec:introduction}
212 
213 For a power series
214 \[
215 F(z)=\sum_{m\geq 0}a_mz^m
216 \]
217 and a positive integer $N$, the coefficients may be separated according to the residue of $m$ modulo $N$.  Root of unity averaging gives exact projection operators for this separation.  This elementary construction is usually called a multisection or a root filter.  It appears throughout the classical theory of generating functions and special functions.  Background on multisections, hypergeometric functions, and related coefficient systems may be found in \cite{Rainville1960,AndrewsAskeyRoy1999,Comtet1974}.
218 
219 Papers I and II developed typed seed grammars, exact projector packets, and
220 finite residue-projector algebra, while Paper III treated typed edge operators
221 and exact coordinate transport \cite{PaperI2026,PaperII2026,PaperIII2026}.
222 The root filter used below is the finite cyclic scalar specialization of that
223 earlier projector calculus. The signed companion matrix is likewise a concrete
224 cyclic instance, not a replacement for Paper III's general typed-edge theory.
225 
226 This volume follows the consequences of applying the cyclic projections to the exponential function.  The projected exponentials form a finite differential system.  After one sign is inserted around the derivative cycle, they form a basis for
227 \[
228 y^{(N)}=-y.
229 \]
230 The section with initial value one is
231 \[
232 \Dfun_N(z)=\sum_{n\geq 0}(-1)^n\frac{z^{Nn}}{(Nn)!}.
233 \]
234 Its reciprocal powers produce polynomial coefficient families. Its logarithmic
235 derivative produces a scalar higher Riccati identity. The same reciprocal
236 construction applied to every exponential section gives normalized
237 residue section functions and ordinary Appell polynomial sequences. These
238 cyclic exponential sections and their matrix addition laws belong to the
239 generalized-hyperbolic-function literature
240 \cite{Schwaiger1992,Muldoon2005}. Appell and binomial-type sequences are
241 discussed in \cite{Roman1984,Comtet1974}.
242 
243 The second half of the paper concerns the zeros of $\Dfun_N$.  The finite exponential sum
244 \[
245 \Dfun_N(z)=\frac1N\sum_{\eta^N=-1}e^{\eta z}
246 \]
247 has a regular polygon of exponents.  The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P\'olya and Langer.  Later developments include the work of Moreno and the treatment in Levin's monograph \cite{Polya1920,Langer1931,Moreno1973,Levin1980}. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficiently large zeros in two term strips and prove exponential localization and simplicity \cite{BleherMallison2006}. The regular polygon and conjugation symmetry of the present family specialize that theory to $N$ exact rays. We give a self-contained specialization because its notation is used throughout the product and zeta theory, and we compute the leading displacement term for this family. The resulting counting formulas have explicit leading constants and bounded error.
248 
249 The rotational symmetry is more useful after the change of variable $w=z^N$.  There is an entire function
250 \[
251 \Delta_N(w)=\sum_{n\geq 0}(-1)^n\frac{w^n}{(Nn)!}
252 \]
253 with
254 \[
255 \Dfun_N(z)=\Delta_N(z^N).
256 \]
257 For $N\geq2$, the order of $\Delta_N$ is $1/N$.  Its Hadamard product therefore has genus zero.  The logarithm of this product identifies the transfer tangent coefficients with power sums of the reduced zeros.  The large zero formula also compares the reduced zero zeta with a Hurwitz zeta function.  Standard material on entire functions, canonical products, and exponential sums appears in \cite{Boas1954,Levin1980,Titchmarsh1939}.  We use singularity analysis in the form described in \cite{FlajoletSedgewick2009} and the classical continuation of the Hurwitz zeta function from \cite{Apostol1976}.
258 
259 The cyclic exponential sections and their matrix addition laws have a
260 classical literature. The tools used below include Appell and binomial type
261 sequences, canonical products, singularity analysis, and Fredholm
262 determinants. These tools are also standard. Our aim is to organize this symmetric
263 specialization and link its reciprocal coefficients to reduced-zero data. No
264 historical priority is asserted for the component constructions.
265 
266 
267 
268 \section{A map of the construction}\label{sec:map}
269 
270 \begin{keyidea}
271 The same object is read in different coordinates. Selecting residues makes its Taylor coefficients visible. A signed shift makes the differential system finite. Passing to $w=z^N$ removes rotational repetition from its zeros. The formulas below identify these descriptions under their stated hypotheses.
272 \end{keyidea}
273 
274 Two variables appear throughout the paper.  The variable $z$ carries the differential equation and the rotational symmetry.  The reduced variable
275 \[
276 w=z^N
277 \]
278 records one representative from each rotational orbit.  Confusing these variables hides the main simplification.  The denominator $\Dfun_N$ has order one as a function of $z$, while $\Delta_N$ has order $1/N$ as a function of $w$.
279 
280 The analytic objects are introduced in the following order.  First,
281 \[
282 \Efun_{N,j}(z)
283 =
284 \sum_{n\geq0}\frac{z^{Nn+j}}{(Nn+j)!}
285 \]
286 is the residue $j$ part of the exponential.  Second,
287 \[
288 \Dfun_{N,j}(z)
289 =
290 \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}
291 \]
292 is the signed basis for $y^{(N)}=-y$.  The section $\Dfun_N=\Dfun_{N,0}$ is the denominator used in the reciprocal family
293 \[
294 \Sfun_N^{(r)}=\Dfun_N^{-r}.
295 \]
296 Set $\Tfun_N=-\Dfun_N^{\prime}/\Dfun_N$. Equivalently, $\Tfun_N$ is the logarithmic derivative of $\Sfun_N^{(1)}$.
297 
298 A different reciprocal family comes from normalizing each exponential section at the origin.  The function
299 \[
300 \Cfun_{N,j}(z)
301 =
302 \frac{z^j/j!}{\Efun_{N,j}(z)}
303 \]
304 is a unit series in $z^N$.  Its powers, after multiplication by $e^{xz}$, give Appell polynomials.  The theta family couples a scaled argument with a reciprocal order.
305 
306 The zero theory begins only after these algebraic relations have been established.  The factorization
307 \[
308 \Dfun_N(z)=\Delta_N(z^N)
309 \]
310 turns every orbit of $N$ zeros in the $z$ plane into one zero in the $w$ plane.  The large zeros in the $z$ plane are located by two dominant exponentials.  The zeros in the $w$ plane then give a genus zero product.  The logarithm of that product returns to the coefficient theory through
311 \[
312 \text{zero power sums}
313 =
314 \text{logarithmic coefficients}.
315 \]
316 This equality is the bridge between the first and second halves of the paper.
317 
318 All powers with a complex parameter are first defined as formal series at the origin.  To represent this germ analytically, use a simply connected zero free region containing the origin and the logarithm normalized to vanish there.  On other regions, specify a branch separately.  Integer reciprocal powers are globally meromorphic and do not require a branch choice.
319 
320 The main statements fall into five groups.  The finite algebra model in \cref{sec:finite_algebra} explains the derivative cycles and addition laws through one exponential identity.  The coefficient theory in \cref{sec:reciprocal_powers} gives two complementary recurrences, one from the original denominator and one from its logarithm.  The Appell theory in \cref{sec:residue_section} supplies both formal differential-operator formulas and a convolution law in the space and order parameters.  The zero analysis in \cref{sec:zero_geometry} specializes classical two term exponential-sum theory, uses symmetry to place the uniquely localized large zeros on the rays, and computes a leading correction.  The product theory in \cref{sec:hadamard} gives the standard diagonal Fredholm encoding of the genus zero product.
321 
322 The volume keeps formal and analytic statements separate.  A formal identity needs only the unit constant term of the relevant series.  A holomorphic complex power also needs a zero free domain and a chosen logarithm.  A meromorphic coefficient asymptotic is stated only for positive integral reciprocal order unless an additional branch domain is specified.
323 
324 \clearpage\part{Cyclic resolution and differential transport}
325 \section{Cyclic projection of analytic functions}\label{sec:cyclic_projection}
326 
327 \begin{readerroadmap}
328 The root filter asks which coefficient addresses belong to a packet. It is an exact selector. Multiplication then asks how two selected packets interact. Their residue labels add modulo $N$, so mixed inputs remain part of the product.
329 \end{readerroadmap}
330 \modelfigure{root_filter}{An exact address model for $N=3$. Every displayed exponent belongs to one residue lane. The drawing records labels rather than coefficient magnitudes.}{fig:root-filter}
331 
332 
333 Fix $N\geq1$ and write
334 \[
335 \omega=e^{2\pi i/N}.
336 \]
337 Let $\Omega$ be a disk centered at the origin, with the whole plane allowed, and assume that $\Omega$ is invariant under multiplication by $\omega$.  For a holomorphic function $F$ on $\Omega$, define
338 \[
339 (\Bcal_{N,j}F)(z)
340 =
341 \frac1N\sum_{\ell=0}^{N-1}\omega^{-j\ell}F(\omega^\ell z),
342 \qquad 0\leq j<N.
343 \]
344 The same formula is valid for formal power series.
345 
Statement: Root filter · Go to proof · Read statement and proof side by side346 \begin{theorem}[Root filter]\label{thm:root_filter}
347 If
348 \[
349 F(z)=\sum_{m\geq0}a_mz^m,
350 \]
351 then
352 \[
353 \Bcal_{N,j}F(z)
354 =
355 \sum_{m\equiv j\bmod N}a_mz^m.
356 \]
357 The operators satisfy
358 \[
359 \Bcal_{N,i}\Bcal_{N,j}=\delta_{ij}\Bcal_{N,j}
360 \]
361 and
362 \[
363 \sum_{j=0}^{N-1}\Bcal_{N,j}=I.
364 \]
365 \end{theorem}
366 
Proof block: Root filter · Go to statement · Read statement and proof side by side367 \begin{proof}
368 Substitution of the power series gives
369 \[
370 \Bcal_{N,j}F(z)
371 =
372 \sum_{m\geq0}a_mz^m
373 \left(
374 \frac1N\sum_{\ell=0}^{N-1}\omega^{(m-j)\ell}
375 \right).
376 \]
377 The inner average is one when $m\equiv j\bmod N$ and zero otherwise.  The operator identities follow from the disjoint residue classes.
378 \end{proof}
379 
Statement: Covariance characterization · Go to proof · Read statement and proof side by side380 \begin{proposition}[Covariance characterization]\label{prop:covariance}
381 A holomorphic function $F$ belongs to the range of $\Bcal_{N,j}$ if and only if
382 \[
383 F(\omega z)=\omega^jF(z).
384 \]
385 \end{proposition}
386 
Proof block: Covariance characterization · Go to statement · Read statement and proof side by side387 \begin{proof}
388 Every monomial selected by $\Bcal_{N,j}$ has the stated covariance.  Conversely, write $F=\sum_k\Bcal_{N,k}F$.  Replacing $z$ by $\omega z$ gives
389 \[
390 \sum_k\omega^k\Bcal_{N,k}F
391 =
392 \omega^j\sum_k\Bcal_{N,k}F.
393 \]
394 Application of $\Bcal_{N,k}$ gives
395 \[
396 (\omega^k-\omega^j)\Bcal_{N,k}F=0.
397 \]
398 The powers $1,\omega,\ldots,\omega^{N-1}$ are distinct, so
399 $\Bcal_{N,k}F=0$ for $k\ne j$.
400 \end{proof}
401 
402 Multiplication is not diagonal in these coordinates. The residue labels add
403 modulo $N$. This is the scalar cyclic instance of the packet product from
404 Paper I.
405 
Statement: Cyclic product law · Go to proof · Read statement and proof side by side406 \begin{theorem}[Cyclic product law]\label{thm:cyclic_product}
407 If $F$ and $G$ are holomorphic on the same rotation-invariant domain, then
408 \[
409 \Bcal_{N,k}(FG)
410 =
411 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
412 (\Bcal_{N,i}F)(\Bcal_{N,j}G).
413 \]
414 \end{theorem}
415 
Proof block: Cyclic product law · Go to statement · Read statement and proof side by side416 \begin{proof}
417 Write $F=\sum_iF_i$ and $G=\sum_jG_j$, where
418 $F_i=\Bcal_{N,i}F$ and $G_j=\Bcal_{N,j}G$.  Their covariance gives
419 \[
420 (F_iG_j)(\omega z)=\omega^{i+j}F_i(z)G_j(z),
421 \]
422 so $F_iG_j$ belongs to the packet labelled by $i+j$ modulo $N$.
423 Expanding the finite product and applying $\Bcal_{N,k}$ gives the formula.
424 \end{proof}
425 
426 \begin{remark}\label{rem:formal_analytic}
427 Every identity in this section holds formally.  In the holomorphic setting, the finite projector sums preserve local uniform convergence.  No choice of branch is involved.
428 \end{remark}
429 
430 \section{The transfer exponential basis}\label{sec:transfer_exponential}
431 
Statement: Definition: def:transfer_exponential432 \begin{definition}\label{def:transfer_exponential}
433 For $\lambda\in\CC$ and $0\leq j<N$, define
434 \[
435 \Efun_{N,j}(\lambda z)
436 =
437 \Bcal_{N,j}(e^{\lambda z})
438 =
439 \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
440 \]
441 \end{definition}
442 
Statement: Differential cycle and initial data · Go to proof · Read statement and proof side by side443 \begin{theorem}[Differential cycle and initial data]\label{thm:exponential_cycle}
444 For $1\leq j<N$,
445 \[
446 \frac{d}{dz}\Efun_{N,j}(\lambda z)
447 =
448 \lambda\Efun_{N,j-1}(\lambda z),
449 \]
450 while
451 \[
452 \frac{d}{dz}\Efun_{N,0}(\lambda z)
453 =
454 \lambda\Efun_{N,N-1}(\lambda z).
455 \]
456 Thus
457 \[
458 \frac{d^N}{dz^N}\Efun_{N,j}(\lambda z)
459 =
460 \lambda^N\Efun_{N,j}(\lambda z).
461 \]
462 For $0\leq k<N$,
463 \[
464 \left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0}
465 =
466 \delta_{kj}\lambda^j.
467 \]
468 If $\lambda\ne0$, the $N$ functions $\Efun_{N,j}(\lambda z)$ form a basis of the solution space of $y^{(N)}=\lambda^Ny$.
469 \end{theorem}
470 
Proof block: Differential cycle and initial data · Go to statement · Read statement and proof side by side471 \begin{proof}
472 Termwise differentiation gives the cyclic derivative formulas.  One full cycle contributes $\lambda^N$.  The first nonzero term of $\Efun_{N,j}(\lambda z)$ is $(\lambda z)^j/j!$, which gives the initial data.  The initial value vectors are linearly independent when $\lambda\ne0$.  The solution space of an order $N$ constant coefficient equation has dimension $N$, so these functions form a basis.
473 \end{proof}
474 
475 
Statement: Wronskian of the transfer basis · Go to proof · Read statement and proof side by side476 \begin{corollary}[Wronskian of the transfer basis]\label{cor:wronskians}
477 Assume $N\geq2$. For $\lambda\ne0$, the Wronskian of the ordered basis
478 \[
479 \Efun_{N,0}(\lambda z),\ldots,\Efun_{N,N-1}(\lambda z)
480 \]
481 is the constant
482 \[
483 \lambda^{N(N-1)/2}.
484 \]
485 \end{corollary}
486 
Proof block: Wronskian of the transfer basis · Go to statement · Read statement and proof side by side487 \begin{proof}
488 At the origin, the derivative matrix is diagonal with entries
489 \[
490 1,\lambda,\ldots,\lambda^{N-1}.
491 \]
492 For an explicit determinant proof, write each basis function as a finite
493 sum of exponentials with exponents $a_\ell=\lambda\omega^\ell$.
494 Its derivative matrix factors as
495 \[
496 W(z)=V\,\operatorname{diag}(e^{a_\ell z})\,C,
497 \qquad V_{k\ell}=a_\ell^k,
498 \]
499 where $C$ is the constant matrix of Fourier weights. Thus
500 \[
501 \det W(z)=e^{z\sum_\ell a_\ell}\det W(0)=\det W(0),
502 \]
503 because $N\geq2$ implies $\sum_\ell\omega^\ell=0$.
504 The product of the diagonal initial values is the stated power of $\lambda$.
505 \end{proof}
506 
Statement: Addition law · Go to proof · Read statement and proof side by side507 \begin{corollary}[Addition law]\label{cor:exponential_addition}
508 For $0\leq k<N$,
509 \[
510 \Efun_{N,k}(\lambda(z+w))
511 =
512 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
513 \Efun_{N,i}(\lambda z)\Efun_{N,j}(\lambda w).
514 \]
515 \end{corollary}
516 
Proof block: Addition law · Go to statement · Read statement and proof side by side517 \begin{proof}
518 Expand $e^{\lambda(z+w)}=e^{\lambda z}e^{\lambda w}$ as a
519 bivariate power series.  A term $z^a w^b$ contributes to the
520 $k$th section exactly when $a+b\equiv k\bmod N$.  Grouping first by
521 $i\equiv a\bmod N$ and $j\equiv b\bmod N$ gives the displayed sum.
522 \end{proof}
523 
524 The basis also has a scalar hypergeometric form.
525 
Statement: Hypergeometric form · Go to proof · Read statement and proof side by side526 \begin{theorem}[Hypergeometric form]\label{thm:hypergeometric_form}
527 Let
528 \[
529 A_{N,j}
530 =
531 \left\{
532 \frac{j+m}{N}
533 \mathrel{:}
534 1\leq m\leq N
535 \text{ and }
536 j+m\ne N
537 \right\}.
538 \]
539 Then
540 \[
541 \Efun_{N,j}(\lambda z)
542 =
543 \frac{(\lambda z)^j}{j!}
544 {}_0F_{N-1}
545 \left(
546 \begin{matrix}-\\ A_{N,j}\end{matrix}
547 \mathrel{,}
548 \frac{(\lambda z)^N}{N^N}
549 \right).
550 \]
551 \end{theorem}
552 
Proof block: Hypergeometric form · Go to statement · Read statement and proof side by side553 \begin{proof}
554 The multiplication formula for the gamma function gives
555 \[
556 (Nn+j)!
557 =
558 j!N^{Nn}n!
559 \prod_{a\in A_{N,j}}(a)_n.
560 \]
561 Indeed, the $N$ Pochhammer factors supplied by the multiplication formula
562 are $((j+m)/N)_n$ for $1\leq m\leq N$; the excluded factor is
563 $((j+N-j)/N)_n=(1)_n=n!$.
564 Insertion into the defining series gives the stated hypergeometric expansion.
565 \end{proof}
566 
567 \begin{example}\label{ex:classical_exponential}
568 For $N=1$, the only section is $e^{\lambda z}$.  For $N=2$,
569 \[
570 \Efun_{2,0}(z)=\cosh z,
571 \qquad
572 \Efun_{2,1}(z)=\sinh z.
573 \]
574 These are classical instances of the same cyclic basis.
575 \end{example}
576 
577 \section{Signed cyclic sections and the distinguished denominator}\label{sec:signed_system}
578 
Statement: Definition: def:signed_sections579 \begin{definition}\label{def:signed_sections}
580 For $0\leq j<N$, define
581 \[
582 \Dfun_{N,j}(z)
583 =
584 \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}.
585 \]
586 The distinguished signed denominator is
587 \[
588 \Dfun_N(z)=\Dfun_{N,0}(z).
589 \]
590 \end{definition}
591 
592 Fix any $\rho\in\CC$ with $\rho^N=-1$.
593 
Statement: Relation with the transfer exponential · Go to proof · Read statement and proof side by side594 \begin{proposition}[Relation with the transfer exponential]\label{prop:signed_exponential_relation}
595 For every $j$,
596 \[
597 \Dfun_{N,j}(z)
598 =
599 \rho^{-j}\Efun_{N,j}(\rho z).
600 \]
601 \end{proposition}
602 
Proof block: Relation with the transfer exponential · Go to statement · Read statement and proof side by side603 \begin{proof}
604 The coefficient of $z^{Nn+j}$ on the right is
605 \[
606 \rho^{-j}\rho^{Nn+j}\frac1{(Nn+j)!}
607 =
608 (-1)^n\frac1{(Nn+j)!}.
609 \]
610 \end{proof}
611 
Statement: Signed derivative cycle · Go to proof · Read statement and proof side by side612 \begin{theorem}[Signed derivative cycle]\label{thm:signed_cycle}
613 For $1\leq j<N$,
614 \[
615 \Dfun_{N,j}'=\Dfun_{N,j-1},
616 \]
617 and
618 \[
619 \Dfun_{N,0}'=-\Dfun_{N,N-1}.
620 \]
621 Every signed section satisfies
622 \[
623 \Dfun_{N,j}^{(N)}=-\Dfun_{N,j}.
624 \]
625 The $N$ signed sections form the basis of $y^{(N)}=-y$ determined by
626 \[
627 \Dfun_{N,j}^{(k)}(0)=\delta_{jk}
628 \]
629 for $0\leq j,k<N$.
630 \end{theorem}
631 
Proof block: Signed derivative cycle · Go to statement · Read statement and proof side by side632 \begin{proof}
633 The derivative formulas follow directly from the series.  The first nonzero term of $\Dfun_{N,j}$ is $z^j/j!$, so the initial value matrix is the identity.  The basis statement follows from the dimension of the solution space.
634 \end{proof}
635 
Statement: Wronskian of the signed basis · Go to proof · Read statement and proof side by side636 \begin{corollary}[Wronskian of the signed basis]\label{cor:signed_wronskian}
637 Assume $N\geq2$. The Wronskian of
638 \[
639 \Dfun_{N,0}(z),\ldots,\Dfun_{N,N-1}(z)
640 \]
641 is one.
642 \end{corollary}
643 
Proof block: Wronskian of the signed basis · Go to statement · Read statement and proof side by side644 \begin{proof}
645 The derivative matrix at the origin is the identity. The determinant
646 factorization in the proof of \cref{cor:wronskians} applies with
647 $a_\ell=\rho\omega^\ell$ and with the signed-section Fourier weights.
648 Since $\sum_\ell a_\ell=0$, the determinant is constant and equals one.
649 \end{proof}
650 
651 
Statement: Twisted addition law · Go to proof · Read statement and proof side by side652 \begin{proposition}[Twisted addition law]\label{prop:signed_addition}
653 For $0\leq k<N$,
654 \[
655 \Dfun_{N,k}(z+w)
656 =
657 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
658 (-1)^{\lfloor(i+j)/N\rfloor}
659 \Dfun_{N,i}(z)\Dfun_{N,j}(w).
660 \]
661 \end{proposition}
662 
Proof block: Twisted addition law · Go to statement · Read statement and proof side by side663 \begin{proof}
664 Use \cref{prop:signed_exponential_relation,cor:exponential_addition}.  The factor introduced by a pair $(i,j)$ is
665 \[
666 \rho^{i+j-k}.
667 \]
668 Since $0\leq i+j\leq2N-2$ and $i+j\equiv k\bmod N$, this factor is one when $i+j=k$ and $-1$ when $i+j=k+N$.
669 \end{proof}
670 
Statement: Finite exponential representation · Go to proof · Read statement and proof side by side671 \begin{theorem}[Finite exponential representation]\label{thm:finite_exponential_representation}
672 Let
673 \[
674 \Rcal_N=\{\eta\in\CC\mathrel{:}\eta^N=-1\}.
675 \]
676 Then
677 \[
678 \Dfun_N(z)
679 =
680 \frac1N\sum_{\eta\in\Rcal_N}e^{\eta z}.
681 \]
682 More generally,
683 \[
684 \Dfun_{N,j}(z)
685 =
686 \frac{\rho^{-j}}N
687 \sum_{\ell=0}^{N-1}
688 \omega^{-j\ell}e^{\rho\omega^\ell z}.
689 \]
690 \end{theorem}
691 
Proof block: Finite exponential representation · Go to statement · Read statement and proof side by side692 \begin{proof}
693 The second formula is the root filter expression for $\rho^{-j}\Efun_{N,j}(\rho z)$.  The first is the case $j=0$.
694 \end{proof}
695 
Statement: Definition: def:reduced_denominator696 \begin{definition}\label{def:reduced_denominator}
697 Define the reduced denominator
698 \[
699 \Delta_N(w)
700 =
701 \sum_{n\geq0}(-1)^n\frac{w^n}{(Nn)!}.
702 \]
703 Then
704 \[
705 \Dfun_N(z)=\Delta_N(z^N).
706 \]
707 \end{definition}
708 
Statement: Symmetries · Go to proof · Read statement and proof side by side709 \begin{corollary}[Symmetries]\label{cor:denominator_symmetries}
710 The denominator satisfies
711 \[
712 \Dfun_N(\omega z)=\Dfun_N(z)
713 \]
714 and
715 \[
716 \Dfun_N(\overline z)=\overline{\Dfun_N(z)}.
717 \]
718 Its nonzero zeros occur in orbits under multiplication by the $N$th roots of unity and under complex conjugation.
719 \end{corollary}
720 
Proof block: Symmetries · Go to statement · Read statement and proof side by side721 \begin{proof}
722 The first identity follows from the powers $z^{Nn}$.  The second follows from the real coefficients.  Both symmetries preserve the zero set and multiplicity.
723 \end{proof}
724 
725 \begin{example}\label{ex:small_denominators}
726 The first denominators are
727 \[
728 \Dfun_1(z)=e^{-z},
729 \qquad
730 \Dfun_2(z)=\cos z.
731 \]
732 For $N=3$,
733 \[
734 \Dfun_3(z)
735 =
736 \frac13\left(
737 e^{-z}
738 +2e^{z/2}\cos\left(\frac{\sqrt3z}{2}\right)
739 \right).
740 \]
741 For $N=4$,
742 \[
743 \Dfun_4(z)
744 =
745 \cos\left(\frac z{\sqrt2}\right)
746 \cosh\left(\frac z{\sqrt2}\right).
747 \]
748 \end{example}
749 
750 \section{A finite algebra and matrix model}\label{sec:finite_algebra}
751 
752 \begin{keyidea}
753 The signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression.
754 \end{keyidea}
755 \modelfigure{signed_cycle}{The exact signed shift $J_4$. Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies $J_4^4=-I$.}{fig:signed-cycle}
756 
757 
758 The derivative and addition laws can be read from one finite algebra identity.  This formulation also identifies the exponent polygon as the spectrum of a signed cyclic shift.
759 
Statement: Definition: def:cyclic_algebras760 \begin{definition}\label{def:cyclic_algebras}
761 Let
762 \[
763 \mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
764 \]
765 and
766 \[
767 \mathfrak A_N^-=\CC[\xi]/(\xi^N+1).
768 \]
769 Every element has a unique representative of degree below $N$.
770 \end{definition}
771 
Statement: Finite algebra exponential identities · Go to proof · Read statement and proof side by side772 \begin{theorem}[Finite algebra exponential identities]\label{thm:finite_algebra_exponentials}
773 In $\mathfrak A_N^+$ one has
774 \[
775 e^{\lambda z\xi}
776 =
777 \sum_{j=0}^{N-1}\Efun_{N,j}(\lambda z)\xi^j.
778 \]
779 In $\mathfrak A_N^-$ one has
780 \[
781 e^{z\xi}
782 =
783 \sum_{j=0}^{N-1}\Dfun_{N,j}(z)\xi^j.
784 \]
785 Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.
786 \end{theorem}
787 
Proof block: Finite algebra exponential identities · Go to statement · Read statement and proof side by side788 \begin{proof}
789 Expand the exponential and write every exponent uniquely as $Nn+j$.  In $\mathfrak A_N^+$, the relation $\xi^{Nn+j}=\xi^j$ gives the transfer sections.  In $\mathfrak A_N^-$, the relation $\xi^{Nn+j}=(-1)^n\xi^j$ gives the signed sections.
790 
791 Differentiation multiplies by $\lambda\xi$ in the first algebra and by $\xi$ in the second.  Reduction of the product by the defining relation gives the derivative cycles.  The identity
792 \[
793 e^{(z+w)\xi}=e^{z\xi}e^{w\xi}
794 \]
795 gives the addition laws after comparison of the coefficients of $1,\xi,\ldots,\xi^{N-1}$.
796 \end{proof}
797 
Statement: Definition: def:signed_shift_matrix798 \begin{definition}\label{def:signed_shift_matrix}
799 Let $J_N$ be the signed cyclic shift on the standard basis $e_0,\ldots,e_{N-1}$ of $\CC^N$, defined by
800 \[
801 J_Ne_j=e_{j+1}
802 \quad (0\leq j<N-1),
803 \qquad
804 J_Ne_{N-1}=-e_0.
805 \]
806 Then $J_N^N=-I$.
807 \end{definition}
808 
Statement: Companion matrix realization · Go to proof · Read statement and proof side by side809 \begin{theorem}[Companion matrix realization]\label{thm:companion_matrix_realization}
810 Put
811 \[
812 \mathbf D_N(z)
813 =
814 \begin{pmatrix}
815 \Dfun_{N,0}(z)\\
816 \Dfun_{N,1}(z)\\
817 \vdots\\
818 \Dfun_{N,N-1}(z)
819 \end{pmatrix}.
820 \]
821 Then
822 \[
823 \mathbf D_N(z)=e^{zJ_N}e_0
824 \]
825 and
826 \[
827 \Dfun_N(z)=\frac1N\tr(e^{zJ_N}).
828 \]
829 The eigenvalues of $J_N$ are the $N$ roots of $-1$, each with multiplicity one.
830 \end{theorem}
831 
Proof block: Companion matrix realization · Go to statement · Read statement and proof side by side832 \begin{proof}
833 The vector $e^{zJ_N}e_0$ has derivative $J_Ne^{zJ_N}e_0$ and initial value $e_0$.  Its components therefore satisfy the same signed derivative cycle and initial data as $\mathbf D_N$.  Uniqueness for the linear system gives the first identity.
834 
835 The vectors
836 \[
837 e_0,J_Ne_0,\ldots,J_N^{N-1}e_0
838 \]
839 are the standard basis.  Thus $e_0$ is cyclic for $J_N$, so the
840 minimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that
841 minimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.
842 The characteristic polynomial has the same degree and is divisible
843 by the minimal polynomial, so it too equals $x^N+1$.  This polynomial
844 has distinct roots.  Consequently $J_N$ is diagonalizable and its
845 spectrum consists of all $N$ roots of $-1$, each once.  Hence
846 \[
847 \tr(e^{zJ_N})
848 =
849 \sum_{\eta^N=-1}e^{\eta z}
850 =
851 N\Dfun_N(z)
852 \]
853 by \cref{thm:finite_exponential_representation}.
854 \end{proof}
855 
Statement: Entries of the fundamental matrix · Go to proof · Read statement and proof side by side856 \begin{corollary}[Entries of the fundamental matrix]\label{cor:fundamental_matrix_entries}
857 For $0\leq p,q<N$, let $d_{pq}$ be the member of $\{0,\ldots,N-1\}$ congruent to $p-q$ modulo $N$.  Put
858 \[
859 \varepsilon_{pq}
860 =
861 \begin{cases}
862 1,&p\geq q,\\
863 -1,&p<q.
864 \end{cases}
865 \]
866 Then
867 \[
868 \bigl(e^{zJ_N}\bigr)_{pq}
869 =
870 \varepsilon_{pq}\Dfun_{N,d_{pq}}(z).
871 \]
872 Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from $q$ to $p$ crosses the signed edge.
873 \end{corollary}
874 
Proof block: Entries of the fundamental matrix · Go to statement · Read statement and proof side by side875 \begin{proof}
876 For every nonnegative integer $n$,
877 \[
878 J_N^{Nn+d_{pq}}e_q
879 =
880 (-1)^n\varepsilon_{pq}e_p.
881 \]
882 Only powers whose exponent is congruent to $d_{pq}$ modulo $N$ contribute to the $(p,q)$ entry of the exponential.  Therefore
883 \[
884 \bigl(e^{zJ_N}\bigr)_{pq}
885 =
886 \varepsilon_{pq}
887 \sum_{n\geq0}(-1)^n
888 \frac{z^{Nn+d_{pq}}}{(Nn+d_{pq})!}.
889 \]
890 This is the stated formula.
891 \end{proof}
892 
Statement: Addition and inversion laws for the companion matrix · Go to proof · Read statement and proof side by side893 \begin{corollary}[Addition and inversion laws for the companion matrix]\label{cor:companion_addition_inversion}
894 Assume $N\geq2$.  The matrix
895 \[
896 U_N(z)=e^{zJ_N}
897 \]
898 satisfies
899 \[
900 U_N(z+w)=U_N(z)U_N(w),
901 \qquad
902 U_N(z)^{-1}=U_N(-z),
903 \qquad
904 \det U_N(z)=1.
905 \]
906 Its first column is $\mathbf D_N(z)$, and all of its entries are given by \cref{cor:fundamental_matrix_entries}.
907 \end{corollary}
908 
Proof block: Addition and inversion laws for the companion matrix · Go to statement · Read statement and proof side by side909 \begin{proof}
910 The first two identities are the exponential group law.  Since $\tr J_N=0$,
911 \[
912 \det(e^{zJ_N})=e^{z\tr J_N}=1.
913 \]
914 The first column statement is \cref{thm:companion_matrix_realization}.
915 \end{proof}
916 
917 \clearpage\part{Reciprocal coefficients and polynomial families}
918 \section{Reciprocal powers and parameter polynomials}\label{sec:reciprocal_powers}
919 
920 \begin{readerroadmap}
921 A unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set.
922 \end{readerroadmap}
923 \modelfigure{reciprocal_normalization}{At $N=2$ and $r=1$, the reciprocal is $\sec z$. The upper row gives block exponential coefficients and the lower row gives ordinary Taylor coefficients. Both rows encode the same five terms.}{fig:reciprocal-normalization}
924 
925 
926 Since $\Dfun_N(0)=1$, its formal logarithm is defined at the origin.  On a
927 simply connected zero free region containing the origin, there is a unique
928 holomorphic branch normalized by $\log\Dfun_N(0)=0$, and its germ at the
929 origin agrees with the formal logarithm.
930 
Statement: Definition: def:reciprocal_family931 \begin{definition}\label{def:reciprocal_family}
932 For $r\in\CC$, define
933 \[
934 \Sfun_N^{(r)}(z)
935 =
936 \Dfun_N(z)^{-r}
937 =
938 \exp\bigl(-r\log\Dfun_N(z)\bigr).
939 \]
940 Write
941 \[
942 \Sfun_N^{(r)}(z)
943 =
944 \sum_{n\geq0}P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}.
945 \]
946 Define the logarithmic coefficients $\kappa_{N,m}$ by
947 \[
948 -\log\Delta_N(w)
949 =
950 \sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}.
951 \]
952 \end{definition}
953 
954 The polynomial dependence on $r$ can be read directly from the formal logarithm.  We use the complete exponential Bell polynomials $B_n$ defined by
955 \[
956 \exp\left(\sum_{m\geq1}x_m\frac{t^m}{m!}\right)
957 =
958 \sum_{n\geq0}B_n(x_1,\ldots,x_n)\frac{t^n}{n!}.
959 \]
960 
Statement: Bell polynomial formula · Go to proof · Read statement and proof side by side961 \begin{theorem}[Bell polynomial formula]\label{thm:reciprocal_bell}
962 For every $n\geq0$,
963 \[
964 P_{N,n}(r)
965 =
966 \frac{(Nn)!}{n!}
967 B_n\bigl(
968  r\kappa_{N,1},
969  r\,1!\,\kappa_{N,2},
970  \ldots,
971  r\,(n-1)!\,\kappa_{N,n}
972 \bigr).
973 \]
974 For $n\geq1$, this is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is
975 \[
976 \frac{(Nn)!}{n!(N!)^n}.
977 \]
978 \end{theorem}
979 
Proof block: Bell polynomial formula · Go to statement · Read statement and proof side by side980 \begin{proof}
981 The identity
982 \[
983 \Delta_N(w)^{-r}
984 =
985 \exp\left(
986  r\sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}
987 \right)
988 \]
989 fits the defining generating function for $B_n$ with
990 \[
991 x_m=r(m-1)!\kappa_{N,m}.
992 \]
993 The coefficient of $w^n$ is therefore the displayed Bell polynomial divided by $n!$.  Multiplication by $(Nn)!$ gives $P_{N,n}(r)$.  Since
994 \[
995 \Delta_N(w)=1-\frac{w}{N!}+O(w^2),
996 \]
997 we have $\kappa_{N,1}=1/N!$.  The term $x_1^n$ gives the stated leading coefficient, which is nonzero.
998 \end{proof}
999 
1000 The coefficients also satisfy a triangular recurrence that does not require the logarithmic coefficients to be known first.
1001 
Statement: Parameter recurrence · Go to proof · Read statement and proof side by side1002 \begin{theorem}[Parameter recurrence]\label{thm:parameter_recurrence}
1003 We have $P_{N,0}(r)=1$.  For $n\geq1$,
1004 \[
1005 \begin{aligned}
1006 P_{N,n}(r)
1007 ={}&-
1008 \sum_{k=1}^{n-1}
1009 (-1)^k
1010 \binom{Nn-1}{Nk}
1011 P_{N,n-k}(r)
1012 \\
1013 &-r
1014 \sum_{k=1}^{n}
1015 (-1)^k
1016 \binom{Nn-1}{Nk-1}
1017 P_{N,n-k}(r).
1018 \end{aligned}
1019 \]
1020 \end{theorem}
1021 
Proof block: Parameter recurrence · Go to statement · Read statement and proof side by side1022 \begin{proof}
1023 Set $S=\Dfun_N^{-r}$.  Logarithmic differentiation gives
1024 \[
1025 \Dfun_NS'+r\Dfun_N'S=0.
1026 \]
1027 Taking the coefficient of $z^{Nn-1}/(Nn-1)!$ gives
1028 \[
1029 \sum_{k=0}^{n-1}(-1)^k
1030 \binom{Nn-1}{Nk}P_{N,n-k}(r)
1031 +r\sum_{k=1}^{n}(-1)^k
1032 \binom{Nn-1}{Nk-1}P_{N,n-k}(r)=0.
1033 \]
1034 The term with $k=0$ in the first sum is $P_{N,n}(r)$, and every
1035 other term has a smaller block index.  Rearrangement gives the formula.
1036 \end{proof}
1037 
Statement: Additive parameter law for reciprocal powers · Go to proof · Read statement and proof side by side1038 \begin{theorem}[Additive parameter law for reciprocal powers]\label{thm:reciprocal_parameter_addition}
1039 For all $r,s\in\CC$,
1040 \[
1041 \Sfun_N^{(r)}(z)\Sfun_N^{(s)}(z)
1042 =
1043 \Sfun_N^{(r+s)}(z)
1044 \]
1045 as a formal identity and on every common zero-free region with a fixed logarithm.  Consequently,
1046 \[
1047 P_{N,n}(r+s)
1048 =
1049 \sum_{q=0}^{n}
1050 \binom{Nn}{Nq}
1051 P_{N,q}(r)P_{N,n-q}(s).
1052 \]
1053 \end{theorem}
1054 
Proof block: Additive parameter law for reciprocal powers · Go to statement · Read statement and proof side by side1055 \begin{proof}
1056 Both statements follow by adding the exponents of the same logarithm and then multiplying the block exponential generating series.
1057 \end{proof}
1058 
Statement: A sequence of binomial type · Go to proof · Read statement and proof side by side1059 \begin{corollary}[A sequence of binomial type]\label{cor:binomial_type}
1060 Define
1061 \[
1062 A_{N,n}(r)
1063 =
1064 \frac{n!}{(Nn)!}P_{N,n}(r).
1065 \]
1066 Then
1067 \[
1068 A_{N,n}(r+s)
1069 =
1070 \sum_{q=0}^{n}
1071 \binom nq
1072 A_{N,q}(r)A_{N,n-q}(s).
1073 \]
1074 Thus $(A_{N,n})_{n\geq0}$ is a polynomial sequence of binomial type.
1075 \end{corollary}
1076 
Proof block: A sequence of binomial type · Go to statement · Read statement and proof side by side1077 \begin{proof}
1078 Multiply the coefficient identity in \cref{thm:reciprocal_parameter_addition} by $n!/(Nn)!$ and simplify the factorials.
1079 \end{proof}
1080 
1081 
1082 The recurrence in \cref{thm:parameter_recurrence} uses the coefficients of the original denominator.  A second recurrence uses only the logarithmic coefficients and is often shorter.
1083 
Statement: Logarithmic Newton recurrence · Go to proof · Read statement and proof side by side1084 \begin{theorem}[Logarithmic Newton recurrence]\label{thm:logarithmic_newton_recurrence}
1085 Set
1086 \[
1087 b_{N,n}(r)=\frac{P_{N,n}(r)}{(Nn)!}.
1088 \]
1089 Then $b_{N,0}(r)=1$ and
1090 for every $n\geq1$,
1091 \[
1092 nb_{N,n}(r)
1093 =
1094 r\sum_{k=1}^{n}\kappa_{N,k}b_{N,n-k}(r).
1095 \]
1096 Equivalently,
1097 \[
1098 P_{N,n}(r)
1099 =
1100 \frac r n
1101 \sum_{k=1}^{n}
1102 \frac{(Nn)!}{(N(n-k))!}
1103 \kappa_{N,k}P_{N,n-k}(r).
1104 \]
1105 If $p_k=r\kappa_{N,k}$, then, for $n\geq1$,
1106 \[
1107 b_{N,n}(r)
1108 =
1109 \frac1{n!}
1110 \det
1111 \begin{pmatrix}
1112 p_1&-1&0&\cdots&0\\
1113 p_2&p_1&-2&\cdots&0\\
1114 p_3&p_2&p_1&\ddots&\vdots\\
1115 \vdots&\vdots&\ddots&\ddots&-(n-1)\\
1116 p_n&p_{n-1}&\cdots&p_2&p_1
1117 \end{pmatrix}.
1118 \]
1119 \end{theorem}
1120 
Proof block: Logarithmic Newton recurrence · Go to statement · Read statement and proof side by side1121 \begin{proof}
1122 Let
1123 \[
1124 B_r(w)=\Delta_N(w)^{-r}
1125 =
1126 \sum_{n\geq0}b_{N,n}(r)w^n.
1127 \]
1128 Logarithmic differentiation gives
1129 \[
1130 \frac{B_r'(w)}{B_r(w)}
1131 =
1132 r\sum_{k\geq1}\kappa_{N,k}w^{k-1}.
1133 \]
1134 Comparison of the coefficient of $w^{n-1}$ proves the recurrence.
1135 
1136 Let $D_n$ denote the displayed determinant, with $D_0=1$.  Expansion along the last row gives
1137 \[
1138 D_n
1139 =
1140 \sum_{k=1}^{n}
1141 \frac{(n-1)!}{(n-k)!}p_kD_{n-k}.
1142 \]
1143 Here the cofactor sign is cancelled by the product of the negative
1144 superdiagonal entries $-(n-k+1),\ldots,-(n-1)$.
1145 Thus $D_n/n!$ satisfies the same recurrence and initial value as $b_{N,n}(r)$.
1146 \end{proof}
1147 
Statement: The first three reciprocal polynomials · Go to proof · Read statement and proof side by side1148 \begin{corollary}[The first three reciprocal polynomials]\label{cor:first_reciprocal_polynomials}
1149 For every $N\geq1$,
1150 \[
1151 P_{N,1}(r)=r,
1152 \]
1153 \[
1154 P_{N,2}(r)
1155 =
1156 r\left[
1157 \frac12\binom{2N}{N}(r+1)-1
1158 \right],
1159 \]
1160 and
1161 \[
1162 P_{N,3}(r)
1163 =
1164 r\left[
1165 \frac{(3N)!}{6(N!)^3}(r+1)(r+2)
1166 -
1167 \binom{3N}{N}(r+1)
1168 +1
1169 \right].
1170 \]
1171 \end{corollary}
1172 
Proof block: The first three reciprocal polynomials · Go to statement · Read statement and proof side by side1173 \begin{proof}
1174 The first logarithmic coefficients are
1175 \[
1176 \kappa_{N,1}=\frac1{N!},
1177 \]
1178 \[
1179 \kappa_{N,2}=\frac1{(N!)^2}-\frac2{(2N)!},
1180 \]
1181 and
1182 \[
1183 \kappa_{N,3}
1184 =
1185 \frac1{(N!)^3}
1186 -
1187 \frac3{N!(2N)!}
1188 +
1189 \frac3{(3N)!}.
1190 \]
1191 Insert these values into \cref{thm:reciprocal_bell} and simplify.
1192 \end{proof}
1193 
Statement: Corollary: cor:factor_r · Go to proof · Read statement and proof side by side1194 \begin{corollary}\label{cor:factor_r}
1195 For $n\geq1$, the polynomial $P_{N,n}(r)$ is divisible by $r$.
1196 \end{corollary}
1197 
Proof block: Corollary: cor:factor_r · Go to statement · Read statement and proof side by side1198 \begin{proof}
1199 At $r=0$, the reciprocal family is the constant function one.  Every positive-index block coefficient therefore vanishes.
1200 \end{proof}
1201 
1202 \begin{example}\label{ex:reciprocal_polynomials}
1203 For $N=1$, one has $\Dfun_1(z)^{-r}=e^{rz}$ and $P_{1,n}(r)=r^n$.  For $N=2$,
1204 \[
1205 P_{2,1}(r)=r,
1206 \]
1207 \[
1208 P_{2,2}(r)=r(3r+2),
1209 \]
1210 and
1211 \[
1212 P_{2,3}(r)=r(15r^2+30r+16).
1213 \]
1214 At $r=1$, these are the positive secant coefficients.
1215 \end{example}
1216 
1217 \section{Logarithmic derivatives and Riccati equations}\label{sec:tangent}
1218 
1219 \begin{microexample}[title={A coefficient calculation at order two}]
1220 Write $\Delta_2(w)^{-r}=\sum_{n\geq0}b_n(r)w^n$. Since
1221 \[
1222 \Delta_2(w)=1-\frac w2+\frac{w^2}{24}+O(w^3),
1223 \qquad -\log\Delta_2(w)=\frac w2+\frac{w^2}{12}+O(w^3),
1224 \]
1225 the first logarithmic coefficients are $\kappa_{2,1}=1/2$ and $\kappa_{2,2}=1/6$. Newton's recurrence gives
1226 \[
1227 b_1(r)=\frac r2,\qquad
1228 2b_2(r)=r\left(\frac r4+\frac16\right).
1229 \]
1230 Restoring the factorial normalization yields $P_{2,1}(r)=r$ and $P_{2,2}(r)=r(3r+2)$. At $r=1$, the second value is five. No zero-location assumption is needed for this finite calculation.
1231 \end{microexample}
1232 \modelfigure{parameter_polynomials}{The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at $r=1$ recover the first two positive-index secant coefficients.}{fig:parameter-polynomials}
1233 
1234 
Statement: Definition: def:transfer_tangent1235 \begin{definition}\label{def:transfer_tangent}
1236 Define the transfer tangent
1237 \[
1238 \Tfun_N(z)
1239 =-\frac{\Dfun_N'(z)}{\Dfun_N(z)}.
1240 \]
1241 Near the origin, its convergent block expansion has the form
1242 \[
1243 \Tfun_N(z)
1244 =
1245 \sum_{n\geq0}T_N(n)
1246 \frac{z^{Nn+N-1}}{(Nn+N-1)!}.
1247 \]
1248 \end{definition}
1249 
Statement: Logarithmic coefficient formula · Go to proof · Read statement and proof side by side1250 \begin{proposition}[Logarithmic coefficient formula]\label{prop:tangent_kappa}
1251 The transfer tangent satisfies
1252 \[
1253 \Tfun_N(z)
1254 =
1255 N\sum_{m\geq1}\kappa_{N,m}z^{Nm-1}.
1256 \]
1257 Equivalently,
1258 \[
1259 T_N(n)
1260 =
1261 N(Nn+N-1)!\kappa_{N,n+1}.
1262 \]
1263 \end{proposition}
1264 
Proof block: Logarithmic coefficient formula · Go to statement · Read statement and proof side by side1265 \begin{proof}
1266 Since $\Dfun_N(z)=\Delta_N(z^N)$,
1267 \[
1268 -\frac{\Dfun_N'(z)}{\Dfun_N(z)}
1269 =
1270 -Nz^{N-1}\frac{\Delta_N'(z^N)}{\Delta_N(z^N)}.
1271 \]
1272 Differentiate the defining series for $-\log\Delta_N$ and compare coefficients.
1273 \end{proof}
1274 
Statement: Tangent recurrence · Go to proof · Read statement and proof side by side1275 \begin{theorem}[Tangent recurrence]\label{thm:tangent_recurrence}
1276 The coefficients satisfy $T_N(0)=1$ and
1277 \[
1278 T_N(n)
1279 =
1280 (-1)^n
1281 -
1282 \sum_{k=1}^{n}
1283 (-1)^k
1284 \binom{Nn+N-1}{Nk}
1285 T_N(n-k).
1286 \]
1287 \end{theorem}
1288 
Proof block: Tangent recurrence · Go to statement · Read statement and proof side by side1289 \begin{proof}
1290 The defining equation is
1291 \[
1292 \Dfun_N\Tfun_N=-\Dfun_N'.
1293 \]
1294 Extracting the coefficient of $z^{Nn+N-1}/(Nn+N-1)!$ gives
1295 \[
1296 \sum_{k=0}^{n}(-1)^k
1297 \binom{Nn+N-1}{Nk}T_N(n-k)=(-1)^n.
1298 \]
1299 Isolating the $k=0$ term gives the recurrence.
1300 \end{proof}
1301 
Statement: Derivative of the reciprocal family · Go to proof · Read statement and proof side by side1302 \begin{corollary}[Derivative of the reciprocal family]\label{cor:reciprocal_derivative}
1303 For every $r\in\CC$, the formal identity
1304 \[
1305 \frac{d}{dz}\Sfun_N^{(r)}(z)
1306 =
1307 r\Tfun_N(z)\Sfun_N^{(r)}(z)
1308 \]
1309 holds.  The same identity holds analytically on every zero-free
1310 domain on which the logarithm used to define $\Sfun_N^{(r)}$ is fixed.
1311 \end{corollary}
1312 
Proof block: Derivative of the reciprocal family · Go to statement · Read statement and proof side by side1313 \begin{proof}
1314 Differentiate $\Dfun_N^{-r}$ on a zero-free region.  The formal identity follows from the same logarithmic calculation.
1315 \end{proof}
1316 
1317 For a function $g$, define the complete differential Bell polynomial $Y_N$ by
1318 \[
1319 \frac{d^N}{dz^N}e^{g(z)}
1320 =
1321 e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr).
1322 \]
1323 
Statement: Scalar higher Riccati identity · Go to proof · Read statement and proof side by side1324 \begin{theorem}[Scalar higher Riccati identity]\label{thm:higher_riccati}
1325 At every point of the open set $\{z:\Dfun_N(z)\ne0\}$,
1326 \[
1327 Y_N\bigl(
1328 -\Tfun_N,
1329 -\Tfun_N',
1330 \ldots,
1331 -\Tfun_N^{(N-1)}
1332 \bigr)
1333 =-1.
1334 \]
1335 \end{theorem}
1336 
Proof block: Scalar higher Riccati identity · Go to statement · Read statement and proof side by side1337 \begin{proof}
1338 Write $\Dfun_N=e^g$ locally.  Then $g'=-\Tfun_N$.  Divide the equation $\Dfun_N^{(N)}=-\Dfun_N$ by $\Dfun_N$ and use the defining identity for $Y_N$.
1339 \end{proof}
1340 
1341 This scalar identity follows from one ordinary differential equation. It does
1342 not redevelop the general nonlinear operation-tree theory of Paper III.
1343 
1344 \begin{example}\label{ex:ordinary_tangent}
1345 For $N=1$, $\Tfun_1=1$.  For $N=2$, $\Dfun_2=\cos z$ and $\Tfun_2=\tan z$.  The Riccati equation becomes
1346 \[
1347 \Tfun_2'=1+\Tfun_2^2.
1348 \]
1349 \end{example}
1350 
1351 \section{Residue-section reciprocals and Appell systems}\label{sec:residue_section}
1352 
1353 The section $\Efun_{N,j}$ has a forced zero of order $j$ at the origin.  Removing it leaves a unit series in $z^N$.
1354 
Statement: Definition: def:normalized_section1355 \begin{definition}\label{def:normalized_section}
1356 For $0\leq j<N$, define
1357 \[
1358 \Phi_{N,j}(w)
1359 =
1360 j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!}.
1361 \]
1362 Then
1363 \[
1364 \Efun_{N,j}(z)
1365 =
1366 \frac{z^j}{j!}\Phi_{N,j}(z^N).
1367 \]
1368 Define the residue-section reciprocal function
1369 \[
1370 \Cfun_{N,j}(z)
1371 =
1372 \Phi_{N,j}(z^N)^{-1}
1373 =
1374 \frac{z^j/j!}{\Efun_{N,j}(z)}.
1375 \]
1376 For $j>0$, the last quotient at $z=0$ is interpreted by its removable
1377 continuation, namely the first expression in the display.
1378 Write
1379 \[
1380 \Cfun_{N,j}(z)
1381 =
1382 \sum_{n\geq0}C_{N,j}(n)\frac{z^{Nn}}{(Nn)!}.
1383 \]
1384 \end{definition}
1385 
Statement: Triangular inverse system · Go to proof · Read statement and proof side by side1386 \begin{theorem}[Triangular inverse system]\label{thm:triangular_inverse}
1387 The coefficients are determined by $C_{N,j}(0)=1$ and
1388 \[
1389 \sum_{q=0}^{n}
1390 \binom{Nn+j}{Nq}
1391 C_{N,j}(q)
1392 =
1393 \delta_{n0}.
1394 \]
1395 For $n\geq1$,
1396 \[
1397 C_{N,j}(n)
1398 =
1399 -
1400 \frac{
1401 \displaystyle\sum_{q=0}^{n-1}
1402 \binom{Nn+j}{Nq}C_{N,j}(q)
1403 }{
1404 \displaystyle\binom{Nn+j}{Nn}
1405 }.
1406 \]
1407 \end{theorem}
1408 
Proof block: Triangular inverse system · Go to statement · Read statement and proof side by side1409 \begin{proof}
1410 Multiply the defining series for $\Cfun_{N,j}$ and $\Efun_{N,j}$.  Their product is $z^j/j!$.  At degree $Nn+j$, multiplication by $(Nn+j)!$ gives the stated binomial coefficient.  The diagonal coefficient is nonzero, so the system is triangular.
1411 \end{proof}
1412 
Statement: Definition: def:residue_section_power1413 \begin{definition}\label{def:residue_section_power}
1414 For $r\in\CC$, define formal coefficients $C_{N,j,n}(r)$ by
1415 \[
1416 \Cfun_{N,j}(z)^r
1417 =
1418 \sum_{n\geq0}C_{N,j,n}(r)\frac{z^{Nn}}{(Nn)!}.
1419 \]
1420 On a simply connected region containing $0$ and avoiding the zeros of $\Phi_{N,j}(z^N)$, interpret this power as $e^{rL}$, where $L$ is the holomorphic logarithm of $\Cfun_{N,j}$ normalized by $L(0)=0$.
1421 \end{definition}
1422 
Statement: Parameter-addition law for residue-section reciprocals · Go to proof · Read statement and proof side by side1423 \begin{theorem}[Parameter-addition law for residue-section reciprocals]\label{thm:residue_section_parameter_addition}
1424 For each $n\geq1$, the coefficient $C_{N,j,n}(r)$ is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is
1425 \[
1426 \frac{(Nn)!}{n!}
1427 \left(-\frac{j!}{(N+j)!}\right)^n.
1428 \]
1429 It satisfies
1430 \[
1431 C_{N,j,n}(r+s)
1432 =
1433 \sum_{q=0}^{n}
1434 \binom{Nn}{Nq}
1435 C_{N,j,q}(r)C_{N,j,n-q}(s).
1436 \]
1437 In particular,
1438 \[
1439 C_{N,j,n}(1)=C_{N,j}(n).
1440 \]
1441 \end{theorem}
1442 
Proof block: Parameter-addition law for residue-section reciprocals · Go to statement · Read statement and proof side by side1443 \begin{proof}
1444 The unit series $\Phi_{N,j}(w)^{-1}$ has a formal logarithm with zero constant term.  Its linear term is
1445 \[
1446 -\frac{j!}{(N+j)!}w.
1447 \]
1448 Exponentiation after multiplication by $r$ shows that the coefficient of $w^n$ is a polynomial of degree at most $n$.  The $n$th power of the linear logarithmic term gives the displayed nonzero leading coefficient, so the degree is exactly $n$.  Multiplication of the powers with parameters $r$ and $s$ gives the coefficient identity.
1449 More explicitly, before the factorial normalization the coefficient of
1450 $r^nw^n$ is
1451 \[
1452 \frac1{n!}\left(-\frac{j!}{(N+j)!}\right)^n;
1453 \]
1454 thus multiplication by $(Nn)!$ gives the displayed leading coefficient.
1455 \end{proof}
1456 
Statement: Definition: def:block_appell1457 \begin{definition}\label{def:block_appell}
1458 Define polynomials $E_m^{(N,j,r)}(x)$ by
1459 \[
1460 \Cfun_{N,j}(z)^re^{xz}
1461 =
1462 \sum_{m\geq0}E_m^{(N,j,r)}(x)\frac{z^m}{m!}.
1463 \]
1464 \end{definition}
1465 
Statement: Appell laws · Go to proof · Read statement and proof side by side1466 \begin{theorem}[Appell laws]\label{thm:appell_laws}
1467 For $m\geq1$, the polynomials satisfy
1468 \[
1469 \frac{d}{dx}E_m^{(N,j,r)}(x)
1470 =
1471 mE_{m-1}^{(N,j,r)}(x)
1472 \]
1473 and, for $m\geq0$,
1474 \[
1475 E_m^{(N,j,r)}(x+y)
1476 =
1477 \sum_{q=0}^{m}
1478 \binom mq
1479 E_q^{(N,j,r)}(x)y^{m-q}.
1480 \]
1481 \end{theorem}
1482 
Proof block: Appell laws · Go to statement · Read statement and proof side by side1483 \begin{proof}
1484 Differentiate the generating function with respect to $x$ for the first identity.  Multiply it by $e^{yz}$ and compare coefficients for the second.
1485 \end{proof}
1486 
Statement: Explicit Appell polynomial · Go to proof · Read statement and proof side by side1487 \begin{proposition}[Explicit Appell polynomial]\label{prop:explicit_appell}
1488 For every $m\geq0$,
1489 \[
1490 E_m^{(N,j,r)}(x)
1491 =
1492 \sum_{0\leq n\leq m/N}
1493 \binom{m}{Nn}
1494 C_{N,j,n}(r)x^{m-Nn}.
1495 \]
1496 \end{proposition}
1497 
Proof block: Explicit Appell polynomial · Go to statement · Read statement and proof side by side1498 \begin{proof}
1499 Multiply the block series for $\Cfun_{N,j}^r$ by the exponential series and collect the coefficient of total degree $m$.
1500 \end{proof}
1501 
1502 
Statement: Joint Appell convolution in the space and order parameters · Go to proof · Read statement and proof side by side1503 \begin{theorem}[Joint Appell convolution in the space and order parameters]\label{thm:appell_two_parameter_convolution}
1504 For all $r,s,x,y\in\CC$ and $m\geq0$,
1505 \[
1506 E_m^{(N,j,r+s)}(x+y)
1507 =
1508 \sum_{q=0}^{m}
1509 \binom mq
1510 E_q^{(N,j,r)}(x)
1511 E_{m-q}^{(N,j,s)}(y).
1512 \]
1513 \end{theorem}
1514 
Proof block: Joint Appell convolution in the space and order parameters · Go to statement · Read statement and proof side by side1515 \begin{proof}
1516 Multiply the generating functions
1517 \[
1518 \Cfun_{N,j}(z)^re^{xz}
1519 \quad\hbox{and}\quad
1520 \Cfun_{N,j}(z)^se^{yz}.
1521 \]
1522 Their product is the generating function with parameters $r+s$ and $x+y$.  Coefficient comparison gives the formula.
1523 \end{proof}
1524 
Statement: Formal differential-operator representation on polynomials · Go to proof · Read statement and proof side by side1525 \begin{proposition}[Formal differential-operator representation on polynomials]\label{prop:appell_operator_form}
1526 Let a formal series $A(z)=\sum_{q\geq0}a_qz^q/q!$ act on polynomials by
1527 \[
1528 A(\partial_x)f(x)
1529 =
1530 \sum_{q\geq0}\frac{a_q}{q!}\partial_x^qf(x).
1531 \]
1532 The power $r$ is applied to the formal symbol before the substitution $z=\partial_x$.  Then
1533 \[
1534 E_m^{(N,j,r)}(x)
1535 =
1536 \Cfun_{N,j}(\partial_x)^r x^m.
1537 \]
1538 The inverse relation is
1539 \[
1540 x^m
1541 =
1542 \Phi_{N,j}(\partial_x^N)^r
1543 E_m^{(N,j,r)}(x).
1544 \]
1545 Both sums are finite on every polynomial.
1546 \end{proposition}
1547 
Proof block: Formal differential-operator representation on polynomials · Go to statement · Read statement and proof side by side1548 \begin{proof}
1549 Applying $A(\partial_x)$ to $e^{xz}$ multiplies it by $A(z)$.  Taking
1550 $A=\Cfun_{N,j}^r$ gives the first identity by the defining generating
1551 function.  Constant coefficient operators on polynomials satisfy
1552 \[
1553 (AB)(\partial_x)=A(\partial_x)B(\partial_x),
1554 \]
1555 because all derivative sums are finite and powers of $\partial_x$
1556 commute.  The two formal symbols
1557 \[
1558 \Cfun_{N,j}(z)^r=\Phi_{N,j}(z^N)^{-r}
1559 \quad\hbox{and}\quad
1560 \Phi_{N,j}(z^N)^r
1561 \]
1562 multiply to one.  Their substitutions at $z=\partial_x$ are therefore
1563 inverse constant coefficient operators on polynomials, which proves
1564 the second identity.
1565 \end{proof}
1566 
1567 
Statement: Appell symmetry · Go to proof · Read statement and proof side by side1568 \begin{corollary}[Appell symmetry]\label{cor:appell_symmetry}
1569 Each polynomial $E_m^{(N,j,r)}$ is monic of degree $m$, and
1570 \[
1571 E_m^{(N,j,r)}(\omega x)
1572 =
1573 \omega^mE_m^{(N,j,r)}(x).
1574 \]
1575 \end{corollary}
1576 
Proof block: Appell symmetry · Go to statement · Read statement and proof side by side1577 \begin{proof}
1578 The term $n=0$ in \cref{prop:explicit_appell} is $x^m$.  Every other exponent is $m-Nn$, so replacement of $x$ by $\omega x$ multiplies every term by $\omega^m$.
1579 \end{proof}
1580 
1581 \begin{example}\label{ex:euler_boundaries}
1582 For $(N,j)=(2,0)$,
1583 \[
1584 \Cfun_{2,0}(z)=\sech z.
1585 \]
1586 For $(N,j)=(2,1)$,
1587 \[
1588 \Cfun_{2,1}(z)=\frac z{\sinh z}.
1589 \]
1590 The first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then
1591 \[
1592 E_m^{(2,0,1)}(x)
1593 =
1594 2^mE_m\left(\frac{x+1}{2}\right).
1595 \]
1596 If $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has
1597 the exact identification
1598 \[
1599 E_m^{(2,1,1)}(x)
1600 =
1601 2^mB_m\left(\frac{x+1}{2}\right),
1602 \]
1603 obtained by substituting $t=2z$ in
1604 $t e^{yt}/(e^t-1)$.
1605 \end{example}
1606 
1607 \section{Scaled reciprocal coupling and theta polynomials}\label{sec:theta}
1608 
Statement: Definition: def:unsigned_denominator1609 \begin{definition}\label{def:unsigned_denominator}
1610 Set
1611 \[
1612 H_N(z)=\Efun_{N,0}(z)
1613 =
1614 \sum_{n\geq0}\frac{z^{Nn}}{(Nn)!}.
1615 \]
1616 For $u\in\CC$, define $\Thetafun_{N,n}(u)$ by
1617 \[
1618 H_N(uz)^{-1}H_N(z)^{-u}
1619 =
1620 \sum_{n\geq0}\Thetafun_{N,n}(u)\frac{z^{Nn}}{(Nn)!}.
1621 \]
1622 This is first a formal identity at the origin.  Analytically, it is interpreted
1623 on a common simply connected zero-free region with the branch of
1624 $\log H_N$ normalized by $\log H_N(0)=0$.
1625 \end{definition}
1626 
Statement: Theta reconstruction · Go to proof · Read statement and proof side by side1627 \begin{theorem}[Theta reconstruction]\label{thm:theta_reconstruction}
1628 For every $n\geq0$,
1629 \[
1630 \Thetafun_{N,n}(u)
1631 =
1632 (-1)^n
1633 \sum_{q=0}^{n}
1634 \binom{Nn}{Nq}
1635 P_{N,q}(1)u^{Nq}P_{N,n-q}(u).
1636 \]
1637 Thus $\Thetafun_{N,n}$ is a polynomial of degree at most $Nn$.
1638 \end{theorem}
1639 
Proof block: Theta reconstruction · Go to statement · Read statement and proof side by side1640 \begin{proof}
1641 Choose $\rho$ with $\rho^N=-1$.  Then
1642 \[
1643 H_N(z)=\Dfun_N(\rho z).
1644 \]
1645 The coefficient of $z^{Nn}$ in $H_N(z)^{-r}$ is therefore $(-1)^nP_{N,n}(r)/(Nn)!$.  Apply this to $H_N(uz)^{-1}$ and $H_N(z)^{-u}$, then multiply the two block series.
1646 \end{proof}
1647 
Statement: Trivial parameter factors · Go to proof · Read statement and proof side by side1648 \begin{proposition}[Trivial parameter factors]\label{prop:theta_factors}
1649 For $n\geq1$, the polynomial $\Thetafun_{N,n}(u)$ is divisible by $u$.  If $N$ is even, it is also divisible by $u+1$.
1650 \end{proposition}
1651 
Proof block: Trivial parameter factors · Go to statement · Read statement and proof side by side1652 \begin{proof}
1653 At $u=0$, the defining product is one.  Hence every positive-index coefficient vanishes.  If $N$ is even, then $H_N(-z)=H_N(z)$.  At $u=-1$, the product is again one, so every positive-index coefficient vanishes there as well.
1654 \end{proof}
1655 
1656 
Statement: Theta Newton recurrence · Go to proof · Read statement and proof side by side1657 \begin{proposition}[Theta Newton recurrence]\label{prop:theta_newton_recurrence}
1658 Put
1659 \[
1660 \theta_{N,n}(u)=\frac{\Thetafun_{N,n}(u)}{(Nn)!}.
1661 \]
1662 Then $\theta_{N,0}(u)=1$ and
1663 \[
1664 n\theta_{N,n}(u)
1665 =
1666 \sum_{m=1}^{n}
1667 (-1)^m\kappa_{N,m}
1668 \bigl(u^{Nm}+u\bigr)
1669 \theta_{N,n-m}(u).
1670 \]
1671 In particular,
1672 \[
1673 \Thetafun_{N,1}(u)=-(u^N+u).
1674 \]
1675 \end{proposition}
1676 
Proof block: Theta Newton recurrence · Go to statement · Read statement and proof side by side1677 \begin{proof}
1678 Write $w=z^N$.  Since $H_N(z)=\Delta_N(-w)$,
1679 \[
1680 -\log H_N(z)
1681 =
1682 \sum_{m\geq1}(-1)^m\kappa_{N,m}\frac{w^m}{m}.
1683 \]
1684 The logarithm of the theta generating function is therefore
1685 \[
1686 \sum_{m\geq1}
1687 (-1)^m\kappa_{N,m}
1688 \bigl(u^{Nm}+u\bigr)
1689 \frac{w^m}{m}.
1690 \]
1691 Logarithmic differentiation and coefficient comparison give the recurrence.  The case $n=1$ uses $\kappa_{N,1}=1/N!$.
1692 \end{proof}
1693 
1694 \begin{example}\label{ex:theta_even}
1695 For $N=2$,
1696 \[
1697 \Thetafun_{2,1}(u)=-u(u+1)
1698 \]
1699 and
1700 \[
1701 \Thetafun_{2,2}(u)
1702 =
1703 u(u+1)(5u^2+u+2).
1704 \]
1705 The two factors predicted by \cref{prop:theta_factors} are visible.
1706 \end{example}
1707 
1708 \clearpage\part{The geometry of zeros}
1709 \section{Growth and the geometry of the large zeros}\label{sec:zero_geometry}
1710 
1711 \begin{readerroadmap}
1712 The finite exponential representation gives a polygon of exponents. Large zeros can occur where two dominant terms nearly cancel. The proof first localizes those zeros and then uses symmetry and uniqueness to place them on the rays. Finite exceptional zeros remain part of the later products and sums.
1713 \end{readerroadmap}
1714 \modelfigure{zero_orbits}{The exact order-four model. The first three zeros on each of four rays are mapped by $w=z^4$ to three positive reduced zeros. The reduced axis uses a logarithmic scale. This complete order-four example does not classify the finite exceptional set at other orders.}{fig:zero-orbits}
1715 
1716 
1717 We now assume $N\geq2$.  Write
1718 \[
1719 \eta_k=e^{(2k+1)\pi i/N},
1720 \qquad 0\leq k<N.
1721 \]
1722 These are the vertices of a regular polygon and
1723 \[
1724 \Dfun_N(z)=\frac1N\sum_{k=0}^{N-1}e^{\eta_kz}.
1725 \]
1726 
Statement: Order from Taylor coefficients1727 \begin{standardfact}[Order from Taylor coefficients]\label{sf:coefficient_order}
1728 If $f(w)=\sum_{n\geq0}a_nw^n$ is an entire function with infinitely many nonzero coefficients, then
1729 \[
1730 \ord f
1731 =
1732 \limsup_{\substack{n\to\infty\\a_n\ne0}}
1733 \frac{n\log n}{\log(1/|a_n|)}.
1734 \]
1735 Canonical products for entire functions of order below one have genus zero.  We use these standard facts in the form given in \cite{Boas1954,Titchmarsh1939}.
1736 \end{standardfact}
1737 
Statement: Orders and exponential type · Go to proof · Read statement and proof side by side1738 \begin{proposition}[Orders and exponential type]\label{prop:orders}
1739 For $N\geq2$,
1740 \[
1741 \ord\Delta_N=\frac1N.
1742 \]
1743 The function $\Dfun_N$ has order one and exponential type one.
1744 \end{proposition}
1745 
Proof block: Orders and exponential type · Go to statement · Read statement and proof side by side1746 \begin{proof}
1747 The coefficient of $w^n$ in $\Delta_N$ has modulus $1/(Nn)!$.
1748 Stirling's formula and \cref{sf:coefficient_order} give order $1/N$.
1749 Since $M_{f(\,\cdot\,^N)}(r)=M_f(r^N)$ for the maximum modulus,
1750 composition with $z^N$ multiplies the order by $N$ and gives order one
1751 for $\Dfun_N$.
1752 
1753 The finite exponential representation gives
1754 \[
1755 |\Dfun_N(z)|\leq e^{|z|},
1756 \]
1757 so the type is at most one.  Choose a root $\eta_0$ and put $z=t\overline{\eta_0}$ with $t>0$.  The term $e^{\eta_0z}=e^t$ has strictly larger real exponent than every other term.  Hence
1758 \[
1759 \Dfun_N(t\overline{\eta_0})
1760 =
1761 \frac1Ne^t(1+o(1)).
1762 \]
1763 The type is therefore one.
1764 \end{proof}
1765 
1766 
Statement: Definition: def:indicator_polygon1767 \begin{definition}\label{def:indicator_polygon}
1768 For $\theta\in\mathbb R$, define
1769 \[
1770 h_N(\theta)
1771 =
1772 \max_{0\leq k<N}
1773 \cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
1774 \]
1775 The tie directions are
1776 \[
1777 \mathfrak T_N
1778 =
1779 \left\{\frac{2\pi k}{N}:0\leq k<N\right\}
1780 \quad\bmod 2\pi.
1781 \]
1782 \end{definition}
1783 
Statement: Indicator polygon and dominant sectors · Go to proof · Read statement and proof side by side1784 \begin{theorem}[Indicator polygon and dominant sectors]\label{thm:indicator_polygon}
1785 The indicator of $\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\Dfun_N(re^{i\theta})\ne0$:
1786 \[
1787 \limsup_{r\to\infty}
1788 \frac1r\log|\Dfun_N(re^{i\theta})|
1789 =
1790 h_N(\theta).
1791 \]
1792 If $K$ is a compact set of directions disjoint from $\mathfrak T_N$, then there are $\delta>0$ and $R>0$ such that, for each $\theta\in K$, one root $\eta(\theta)$ of $-1$ satisfies
1793 \[
1794 \Dfun_N(re^{i\theta})
1795 =
1796 \frac1N
1797 \exp\bigl(\eta(\theta)re^{i\theta}\bigr)
1798 \left(1+O(e^{-\delta r})\right)
1799 \]
1800 uniformly for $r\geq R$.  In particular, no sufficiently large zero can lie in those directions.
1801 \end{theorem}
1802 
Proof block: Indicator polygon and dominant sectors · Go to statement · Read statement and proof side by side1803 \begin{proof}
1804 For $z=re^{i\theta}$, the real part of the exponent $\eta_kz$ is
1805 \[
1806 r\cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
1807 \]
1808 The triangle inequality gives the upper indicator bound $h_N(\theta)$.
1809 
1810 Away from a tie direction, the maximizing vertex is unique.  It is constant
1811 on each connected component of the complement of the tie directions.  A
1812 compact set avoiding all ties meets only finitely many such components, and
1813 compactness gives one uniform positive lower bound for every relevant
1814 projection gap.  Division by the maximizing exponential gives the stated
1815 uniform asymptotic and the zero-free conclusion.
1816 
1817 At a tie direction exactly two adjacent vertices attain the maximum.  After rotation it is enough to take $\theta=0$.  With $c_N=\cos(\pi/N)$ and $s_N=\sin(\pi/N)$, the two dominant terms give
1818 \[
1819 \frac2N e^{c_Nr}\cos(s_Nr)
1820 \]
1821 and every other term is exponentially smaller.  Along a sequence on which $|\cos(s_Nr)|\geq1/2$, the lower indicator bound is $c_N=h_N(0)$.  Rotational symmetry treats every tie direction.
1822 \end{proof}
1823 
1824 The zero strings are governed by two adjacent exponents.  Put
1825 \[
1826 a_N=\frac\pi N,
1827 \qquad
1828 s_N=\sin a_N,
1829 \qquad
1830 c_N=\cos a_N,
1831 \]
1832 and
1833 \[
1834 A_N=\frac\pi{s_N},
1835 \qquad
1836  t_m=A_N\left(m+\frac12\right).
1837 \]
1838 For $N\geq3$, also put
1839 \[
1840 \gamma_N=c_N-\cos(3a_N)>0.
1841 \]
1842 
Statement: Two term dominance · Go to proof · Read statement and proof side by side1843 \begin{lemma}[Two term dominance]\label{lem:two_term_dominance}
1844 Assume $N\geq3$.  There are positive constants $Y$, $R$, and $C$ with the following properties.
1845 
1846 If $z=x+iy$ lies in the sector $|\arg z|\leq a_N$, has $x\geq R$, and is a zero of $\Dfun_N$, then $|y|\leq Y$.
1847 
1848 On the strip $x\geq R$ and $|y|\leq Y$, define
1849 \[
1850 R_N(z)
1851 =
1852 \sum_{\substack{\eta^N=-1\\
1853 \eta\ne e^{ia_N},e^{-ia_N}}}
1854 \exp\bigl((\eta-e^{-ia_N})z\bigr).
1855 \]
1856 Then
1857 \[
1858 N e^{-e^{-ia_N}z}\Dfun_N(z)
1859 =
1860 1+e^{2is_Nz}+R_N(z),
1861 \]
1862 and, for $q=0,1,2$,
1863 \[
1864 |R_N^{(q)}(z)|
1865 \leq
1866 C e^{-\gamma_Nx}.
1867 \]
1868 \end{lemma}
1869 
Proof block: Two term dominance · Go to statement · Read statement and proof side by side1870 \begin{proof}
1871 A direct projection estimate gives the stronger bound
1872 \[
1873 Y=\frac{\log(N-1)}{2s_N}
1874 \]
1875 for every zero in the sector, without a large-radius assumption.
1876 Write $z=re^{i\theta}$ with $0\leq\theta\leq a_N$ and $r\geq0$.
1877 For $0\leq\ell\leq N-2$, the angle
1878 $\theta+(2\ell+1)a_N$ lies between $a_N+\theta$ and
1879 $2\pi-a_N-\theta$. Monotonicity of cosine on $[0,\pi]$, followed by
1880 reflection about $\pi$, therefore gives
1881 \[
1882 \Re(\eta_\ell z)\leq r\cos(a_N+\theta).
1883 \]
1884 The remaining vertex $e^{-ia_N}$ has projection $r\cos(a_N-\theta)$.
1885 Their difference is at least $2s_Nr\sin\theta=2s_Ny$.
1886 If $\Dfun_N(z)=0$, divide its finite exponential sum by the term
1887 with exponent $e^{-ia_N}z$ and apply the triangle inequality:
1888 \[
1889 1\leq(N-1)e^{-2s_Ny}.
1890 \]
1891 Thus $y\leq Y$. Conjugation gives the same bound for $-y$ on the lower
1892 half sector. Since $N\geq3$ and $s_N>0$, this $Y$ is positive.
1893 
1894 On the resulting strip, the roots $e^{ia_N}$ and $e^{-ia_N}$ have
1895 real part $c_N$, while each remaining root $\eta=u+iv$ satisfies
1896 $u\leq\cos(3a_N)=c_N-\gamma_N$. Hence
1897 \[
1898 \Re\bigl((\eta-e^{-ia_N})z\bigr)
1899 \leq-\gamma_Nx+2Y.
1900 \]
1901 Division by $e^{e^{-ia_N}z}$ gives the displayed normalization.
1902 Each differentiation multiplies a remainder term by
1903 $\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,
1904 \[
1905 |R_N^{(q)}(z)|\leq(N-2)2^qe^{2Y}e^{-\gamma_Nx}.
1906 \]
1907 One may take $R=1$ and $C=4(N-2)e^{2Y}$.
1908 \end{proof}
1909 
Statement: Near cancellation of two exponentials · Go to proof · Read statement and proof side by side1910 \begin{lemma}[Near cancellation of two exponentials]\label{lem:near_cancellation}
1911 For every $M>0$, there are constants $\varepsilon_M>0$ and $C_M>0$ such that the following holds.  If
1912 \[
1913 |\Re w|\leq M
1914 \]
1915 and
1916 \[
1917 |1+e^w|\leq\varepsilon_M,
1918 \]
1919 then there is a unique integer $m$ with
1920 \[
1921 |w-(2m+1)\pi i|
1922 \leq
1923 C_M|1+e^w|.
1924 \]
1925 \end{lemma}
1926 
Proof block: Near cancellation of two exponentials · Go to statement · Read statement and proof side by side1927 \begin{proof}
1928 In fact, the constants $\varepsilon_M=1/2$ and $C_M=3/2$ work
1929 for every complex $w$, without the strip restriction. Put
1930 $u=-(1+e^w)$ and $L=\log(1+u)=\log(-e^w)$, using the principal
1931 logarithm. The elementary logarithm estimate on $|u|\leq1/2$ gives
1932 $|L|\leq(3/2)|u|$. Since $e^{L+\pi i}=e^w$, there is an integer
1933 $m$ such that
1934 \[
1935 w=L+(2m+1)\pi i.
1936 \]
1937 This gives the required distance bound. Distinct centers are separated
1938 by at least $2\pi$, whereas two centers satisfying the bound would
1939 be separated by at most $3|1+e^w|\leq3/2$. Thus the integer is unique.
1940 The formal proof uses the convergent logarithm estimate and the exact
1941 periodicity theorem for the complex exponential.
1942 \end{proof}
1943 
Statement: Large zero strings · Go to proof · Read statement and proof side by side1944 \begin{theorem}[Large zero strings]\label{thm:large_zero_strings}
1945 Assume $N\geq3$.  There are constants $m_0$, $C$, and $\gamma>0$ and a sequence of positive real simple zeros $(\tau_m)_{m\geq m_0}$ of $\Dfun_N$ such that
1946 \[
1947 \tau_m
1948 =
1949  t_m+O(e^{-\gamma t_m}).
1950 \]
1951 Every zero of sufficiently large modulus is of the form
1952 \[
1953 \omega^k\tau_m
1954 \]
1955 for a unique $0\leq k<N$ and a unique $m\geq m_0$.
1956 
1957 The derivatives satisfy
1958 \[
1959 \Dfun_N'(\omega^k\tau_m)
1960 =
1961 \omega^{-k}\Dfun_N'(\tau_m)
1962 \]
1963 and
1964 \[
1965 |\Dfun_N'(\tau_m)|
1966 =
1967 \frac{2s_N}{N}e^{c_N\tau_m}
1968 \left(1+O(e^{-\gamma\tau_m})\right).
1969 \]
1970 \end{theorem}
1971 
Proof block: Large zero strings · Go to statement · Read statement and proof side by side1972 \begin{proof}
1973 Work first in the sector $|\arg z|\leq a_N$.  By \cref{lem:two_term_dominance}, every sufficiently large zero lies in a fixed strip about the positive real axis.  Put
1974 \[
1975 G(z)=N e^{-e^{-ia_N}z}\Dfun_N(z).
1976 \]
1977 Then
1978 \[
1979 G(z)=1+e^{2is_Nz}+R_N(z)
1980 \]
1981 with an exponentially small remainder and derivative.
1982 
1983 If $G(z)=0$, then
1984 \[
1985 |1+e^{2is_Nz}|
1986 \leq
1987 C e^{-\gamma_N\Re z}.
1988 \]
1989 The real part of $2is_Nz$ is bounded in the strip.  Apply
1990 \cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique
1991 integer $m$ such that
1992 \[
1993 |z-t_m|\leq C_1e^{-\gamma_N\Re z},
1994 \]
1995 where
1996 \[
1997  t_m=\frac{(2m+1)\pi}{2s_N}.
1998 \]
1999 In particular, $|\Re z-t_m|=o(1)$.  For all sufficiently large zeros
2000 we have $|\Re z-t_m|\leq1$, and hence
2001 \[
2002 |z-t_m|\leq C_2e^{-\gamma_Nt_m}
2003 \]
2004 with a constant independent of $m$.
2005 Because $\Re z\to+\infty$ in this argument, the corresponding integers
2006 $m$ are eventually nonnegative and tend to infinity.
2007 
2008 Fix a large $m$, put $a=2is_N$, and consider the closed disk
2009 $B_m=\{z:|z-t_m|\leq 1/(2s_N)\}$. The remainder bounds hold
2010 uniformly on these fixed-radius disks. For sufficiently large $m$,
2011 $|R_N|\leq 1/2$ and $|R_N'|\leq s_N/2$ there. Define
2012 \[
2013 \Phi_m(z)=t_m+\frac{\log(1+R_N(z))}{2is_N},
2014 \]
2015 using the principal logarithm. The estimate
2016 $|\log(1+u)|\leq(3/2)|u|$ shows that $\Phi_m$ maps $B_m$ into
2017 itself. The derivative bound $|\Phi_m'|\leq1/2$ makes it a contraction.
2018 Banach's fixed-point theorem gives a unique fixed point.
2019 On $B_m$, $|a(z-t_m)|\leq1<\pi$, so the principal logarithm inverts
2020 the exponential and the fixed-point equation is equivalent to $G(z)=0$.
2021 Moreover,
2022 \[
2023 |z-t_m|\leq\frac{3}{4s_N}|R_N(z)|=O(e^{-\gamma_Nt_m}).
2024 \]
2025 At this zero,
2026 \[
2027 |G'(z)|\geq |a|\,|1+R_N(z)|-|R_N'(z)|\geq s_N/2>0,
2028 \]
2029 so the zero is simple. The disks are mutually disjoint because their
2030 diameter $1/s_N$ is smaller than the center spacing $\pi/s_N$.
2031 Enlarging the strip if necessary, these disks lie inside the fundamental
2032 sector and the fixed strip for large $m$.
2033 
2034 The localization estimate puts every sufficiently large zero in the
2035 fundamental sector into one of these disks.  The zero sets of $G$ and
2036 $\Dfun_N$ coincide because their prefactor is nonzero.  Each disk is
2037 invariant under complex conjugation, and $\Dfun_N$ has real
2038 coefficients.  The conjugate of its unique zero is therefore the same
2039 zero.  It is real, positive for large $m$, and simple by the
2040 derivative bound above.  Denote it by $\tau_m$.
2041 
2042 The parts of the fundamental sector away from the positive tie ray,
2043 including its boundary rays, were already excluded uniformly.
2044 Rotational invariance now gives every remaining large zero as
2045 $\omega^k\tau_m$.  Distinct rays and the disjoint disks give uniqueness
2046 of $k$ and $m$.  Taking $\gamma=\gamma_N$ proves the zero location in
2047 the statement.
2048 
2049 Differentiate the normalized expression.  At $z=\tau_m$,
2050 \[
2051 G'(\tau_m)
2052 =
2053 2is_Ne^{2is_N\tau_m}+R_N'(\tau_m)
2054 =
2055 -2is_N+O(e^{-\gamma_N\tau_m}).
2056 \]
2057 Since $G(\tau_m)=0$,
2058 \[
2059 \Dfun_N'(\tau_m)
2060 =
2061 \frac1N e^{e^{-ia_N}\tau_m}G'(\tau_m).
2062 \]
2063 Taking absolute values gives the derivative asymptotic.  Differentiating $\Dfun_N(\omega z)=\Dfun_N(z)$ gives the rotation formula.
2064 \end{proof}
2065 
Statement: Leading displacement term and derivative sign · Go to proof · Read statement and proof side by side2066 \begin{theorem}[Leading displacement term and derivative sign]\label{thm:first_zero_correction}
2067 With the notation of \cref{lem:two_term_dominance,thm:large_zero_strings},
2068 \[
2069 \tau_m
2070 =
2071 t_m+
2072 \frac{R_N(t_m)}{2is_N}
2073 +
2074 O(e^{-2\gamma_Nt_m}).
2075 \]
2076 The quotient $R_N(t_m)/(2is_N)$ is real.  One also has
2077 \[
2078 \Dfun_N'(\tau_m)
2079 =
2080 (-1)^{m+1}
2081 \frac{2s_N}{N}
2082 e^{c_N\tau_m}
2083 \left(1+O(e^{-\gamma_N\tau_m})\right).
2084 \]
2085 \end{theorem}
2086 
Proof block: Leading displacement term and derivative sign · Go to statement · Read statement and proof side by side2087 \begin{proof}
2088 Put $\delta_m=\tau_m-t_m$.  The large zero theorem gives $\delta_m=O(e^{-\gamma_Nt_m})$.  Since $e^{2is_Nt_m}=-1$,
2089 \[
2090 1+e^{2is_N(t_m+\delta_m)}
2091 =
2092 -2is_N\delta_m+O(\delta_m^2).
2093 \]
2094 The remainder estimate through the first derivative gives
2095 \[
2096 R_N(t_m+\delta_m)
2097 =
2098 R_N(t_m)+O(e^{-\gamma_Nt_m}|\delta_m|).
2099 \]
2100 Substitution in the zero equation gives the displacement formula.
2101 
2102 At the real point $t_m$, the normalized value
2103 \[
2104 N e^{-e^{-ia_N}t_m}\Dfun_N(t_m)
2105 \]
2106 is purely imaginary.  The two dominant terms cancel there, so $R_N(t_m)$ is purely imaginary and the displayed correction is real.
2107 
2108 For the derivative, use the formula in the proof of \cref{thm:large_zero_strings} and
2109 \[
2110 e^{-is_N\tau_m}
2111 =
2112 (-1)^m(-i)
2113 \left(1+O(e^{-\gamma_N\tau_m})\right).
2114 \]
2115 The stated real sign follows.
2116 \end{proof}
2117 
Statement: Leading correction for N=3 · Go to proof · Read statement and proof side by side2118 \begin{corollary}[Leading correction for $N=3$]\label{cor:N3_zero_correction}
2119 For $N=3$,
2120 \[
2121 t_m=\frac{2\pi}{\sqrt3}\left(m+\frac12\right)
2122 \]
2123 and
2124 \[
2125 \tau_m
2126 =
2127 t_m+
2128 \frac{(-1)^m}{\sqrt3}e^{-3t_m/2}
2129 +
2130 O(e^{-3t_m}).
2131 \]
2132 \end{corollary}
2133 
Proof block: Leading correction for N=3 · Go to statement · Read statement and proof side by side2134 \begin{proof}
2135 The only nondominant exponent is $-1$.  Hence
2136 \[
2137 R_3(t)
2138 =
2139 \exp\left(\left(-\frac32+\frac{\sqrt3}{2}i\right)t\right).
2140 \]
2141 At $t=t_m$, its phase is $i(-1)^m$.  Insert this value into \cref{thm:first_zero_correction}.
2142 \end{proof}
2143 
Statement: The case N=2 · Go to proof · Read statement and proof side by side2144 \begin{corollary}[The case $N=2$]\label{cor:case_two_zeros}
2145 For $N=2$,
2146 \[
2147 \Dfun_2(z)=\cos z.
2148 \]
2149 Its zeros are
2150 \[
2151 \pm\left(m+\frac12\right)\pi,
2152 \qquad m\geq0,
2153 \]
2154 and all are simple.  They agree with the formula $t_m=A_2(m+1/2)$.
2155 \end{corollary}
2156 
Proof block: The case N=2 · Go to statement · Read statement and proof side by side2157 \begin{proof}
2158 This is the standard zero set of the cosine function.
2159 \end{proof}
2160 
Statement: Eventual simplicity and exact rays · Go to proof · Read statement and proof side by side2161 \begin{corollary}[Eventual simplicity and exact rays]\label{cor:eventual_simplicity}
2162 For every $N\geq2$, all but finitely many zeros of $\Dfun_N$ are simple.  Every sufficiently large zero lies on one of the rays
2163 \[
2164 \omega^k(0,\infty),
2165 \qquad 0\leq k<N.
2166 \]
2167 \end{corollary}
2168 
Proof block: Eventual simplicity and exact rays · Go to statement · Read statement and proof side by side2169 \begin{proof}
2170 Use \cref{thm:large_zero_strings} for $N\geq3$ and \cref{cor:case_two_zeros} for $N=2$.
2171 \end{proof}
2172 
2173 
2174 \begin{example}[Exact fourth-order strings]\label{ex:fourth_order_zeros}
2175 For $N=4$,
2176 \[
2177 \Dfun_4(z)
2178 =
2179 \cos\left(\frac z{\sqrt2}\right)
2180 \cosh\left(\frac z{\sqrt2}\right).
2181 \]
2182 The positive zeros are exactly
2183 \[
2184 \tau_m
2185 =
2186 \sqrt2\pi\left(m+\frac12\right)
2187 =
2188 A_4\left(m+\frac12\right).
2189 \]
2190 Rotation by the fourth roots of unity gives the real and imaginary zero strings.  In this case the error term in \cref{thm:large_zero_strings} is identically zero.
2191 \end{example}
2192 
Statement: Reduced zeros · Go to proof · Read statement and proof side by side2193 \begin{corollary}[Reduced zeros]\label{cor:reduced_zeros}
2194 Let $\Lambda_N$ be the nonzero zero multiset of $\Delta_N$.  All but finitely many members of $\Lambda_N$ are positive and simple.  They can be indexed as
2195 \[
2196 \lambda_m=\tau_m^N
2197 \]
2198 for all sufficiently large $m$.  If $N\geq3$, the notation may be chosen
2199 as in \cref{thm:large_zero_strings}, and
2200 \[
2201 \lambda_m
2202 =
2203  t_m^N
2204 +O\left(t_m^{N-1}e^{-\gamma t_m}\right).
2205 \]
2206 For $N=2$, take $\tau_m=t_m=\pi(m+1/2)$ for $m\geq0$; then
2207 $\lambda_m=t_m^2$ exactly.
2208 \end{corollary}
2209 
Proof block: Reduced zeros · Go to statement · Read statement and proof side by side2210 \begin{proof}
2211 The identity $\Dfun_N(z)=\Delta_N(z^N)$ identifies each orbit of $N$ nonzero zeros of $\Dfun_N$ with one zero of $\Delta_N$.  For $N\geq3$, the large orbit representatives are the positive zeros $\tau_m$ from \cref{thm:large_zero_strings}; for $N=2$, use \cref{cor:case_two_zeros}.  Simplicity is preserved because
2212 \[
2213 \Dfun_N'(z)=Nz^{N-1}\Delta_N'(z^N)
2214 \]
2215 and $z\ne0$.  The asymptotic follows by expanding $\tau_m^N$ about $t_m$.
2216 \end{proof}
2217 
Statement: Zero counting asymptotics · Go to proof · Read statement and proof side by side2218 \begin{theorem}[Zero counting asymptotics]\label{thm:zero_counting}
2219 Let $n_{\Dfun_N}(R)$ count the zeros of $\Dfun_N$ in $|z|\leq R$, with multiplicity.  Let $n_{\Delta_N}(R)$ count the zeros of $\Delta_N$ in $|w|\leq R$.  Then, as $R\to\infty$,
2220 \[
2221 n_{\Dfun_N}(R)
2222 =
2223 \frac{N\sin(\pi/N)}{\pi}R+O(1)
2224 \]
2225 and
2226 \[
2227 n_{\Delta_N}(R)
2228 =
2229 \frac{\sin(\pi/N)}{\pi}R^{1/N}+O(1).
2230 \]
2231 \end{theorem}
2232 
Proof block: Zero counting asymptotics · Go to statement · Read statement and proof side by side2233 \begin{proof}
2234 Each of the $N$ rays contains one large zero near
2235 \[
2236  t_m=\frac\pi{\sin(\pi/N)}\left(m+\frac12\right).
2237 \]
2238 Since $\tau_m-t_m=o(1)$ and the model spacing is the positive constant
2239 $A_N$, replacing $\tau_m\leq R$ by $t_m\leq R$ changes the count by at
2240 most a bounded amount.
2241 The number of such indices with $t_m\leq R$ is
2242 \[
2243 \frac{\sin(\pi/N)}\pi R+O(1).
2244 \]
2245 Multiplication by $N$ and addition of the finite exceptional set give the first formula.  The second follows from
2246 \[
2247 n_{\Dfun_N}(R^{1/N})=Nn_{\Delta_N}(R).
2248 \]
2249 \end{proof}
2250 
2251 
Statement: Zero exponents · Go to proof · Read statement and proof side by side2252 \begin{corollary}[Zero exponents]\label{cor:exponent_of_convergence}
2253 The exponent of convergence of the nonzero zeros of $\Dfun_N$ is one.  The exponent of convergence of the zeros of $\Delta_N$ is $1/N$.
2254 \end{corollary}
2255 
Proof block: Zero exponents · Go to statement · Read statement and proof side by side2256 \begin{proof}
2257 The asymptotic counting formulas in \cref{thm:zero_counting} give the two exponents by partial summation.
2258 \end{proof}
2259 
2260 \clearpage\part{Products, moments, and zero zeta functions}
2261 \section{Hadamard products and zero power sums}\label{sec:hadamard}
2262 
2263 \begin{keyidea}
2264 The product uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset.
2265 \end{keyidea}
2266 \modelfigure{zero_moments}{Order-two moment sums over the first forty reduced zeros $\lambda_k=\pi^2(k+1/2)^2$. The curves are finite numerical partial sums. The dashed limits $1/2$ and $1/6$ follow from the exact logarithmic coefficients and the product identity.}{fig:zero-moments}
2267 
2268 
2269 The reduction from $z$ to $w=z^N$ removes the rotational repetition of the zeros.
2270 
Statement: Genus zero product · Go to proof · Read statement and proof side by side2271 \begin{theorem}[Genus zero product]\label{thm:genus_zero_product}
2272 For $N\geq2$, let $\Lambda_N$ be the nonzero zero multiset of $\Delta_N$.  Then
2273 \[
2274 \Delta_N(w)
2275 =
2276 \prod_{\lambda\in\Lambda_N}
2277 \left(1-\frac w\lambda\right),
2278 \]
2279 where the product converges locally uniformly.  Consequently,
2280 \[
2281 \Dfun_N(z)
2282 =
2283 \prod_{\lambda\in\Lambda_N}
2284 \left(1-\frac{z^N}{\lambda}\right).
2285 \]
2286 \end{theorem}
2287 
Proof block: Genus zero product · Go to statement · Read statement and proof side by side2288 \begin{proof}
2289 By \cref{prop:orders}, the order of $\Delta_N$ is $1/N<1$.  By
2290 \cref{cor:reduced_zeros}, it has infinitely many zeros, and
2291 \cref{cor:exponent_of_convergence} gives
2292 \[
2293 \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty.
2294 \]
2295 Hadamard factorization therefore gives a canonical product of genus zero.
2296 The only possible exponential factor is a nonzero constant because the order
2297 is below one.  Since $\Delta_N(0)=1$, that constant is one.  Substitution of
2298 $w=z^N$ gives the second formula.
2299 \end{proof}
2300 
2301 
Statement: Diagonal Fredholm determinant representation · Go to proof · Read statement and proof side by side2302 \begin{theorem}[Diagonal Fredholm determinant representation]\label{thm:trace_class_model}
2303 Let $I_N$ be a multiplicity-sensitive countable index set for $\Lambda_N$
2304 and put $\mathcal H_N=\ell^2(I_N)$.  On its standard orthonormal basis,
2305 define the diagonal operator
2306 \[
2307 K_Ne_\lambda=\lambda^{-1}e_\lambda.
2308 \]
2309 Then $K_N$ is trace class and
2310 \[
2311 \Delta_N(w)=\Fdet(I-wK_N).
2312 \]
2313 For every $m\geq1$,
2314 \[
2315 \kappa_{N,m}=\tr(K_N^m).
2316 \]
2317 \end{theorem}
2318 
Proof block: Diagonal Fredholm determinant representation · Go to statement · Read statement and proof side by side2319 \begin{proof}
2320 By \cref{cor:exponent_of_convergence},
2321 \[
2322 \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty.
2323 \]
2324 Thus the diagonal operator $K_N$ is trace class.  Its Fredholm determinant is the locally uniform product
2325 \[
2326 \Fdet(I-wK_N)
2327 =
2328 \prod_{\lambda\in\Lambda_N}
2329 \left(1-\frac w\lambda\right),
2330 \]
2331 which equals $\Delta_N(w)$ by \cref{thm:genus_zero_product}.
2332 
2333 Taking the logarithm gives
2334 \[
2335 -\log\Delta_N(w)
2336 =
2337 \sum_{m\geq1}\frac{w^m}{m}\tr(K_N^m)
2338 \]
2339 near the origin.  Comparison with the definition of $\kappa_{N,m}$ proves the trace identity.
2340 \end{proof}
2341 
2342 This is the standard diagonal encoding of this family's genus zero product.
2343 It is not a general trace-class resolvent or Feshbach theorem.
2344 
Statement: Complete symmetric function form · Go to proof · Read statement and proof side by side2345 \begin{corollary}[Complete symmetric function form]\label{cor:complete_symmetric_form}
2346 For a positive integer $r$, let each eigenvalue $\lambda^{-1}$ of $K_N$ be repeated $r$ times.  If $h_n$ denotes the complete homogeneous symmetric function of degree $n$ in this absolutely summable multiset, then
2347 \[
2348 \frac{P_{N,n}(r)}{(Nn)!}=h_n.
2349 \]
2350 For arbitrary complex $r$, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by $r$.
2351 \end{corollary}
2352 
Proof block: Complete symmetric function form · Go to statement · Read statement and proof side by side2353 \begin{proof}
2354 The generating function for the complete homogeneous symmetric functions is
2355 \[
2356 \prod_{\lambda\in\Lambda_N}
2357 \left(1-\frac w\lambda\right)^{-r}
2358 =
2359 \Delta_N(w)^{-r}.
2360 \]
2361 Coefficient comparison gives the first identity.  Power sums determine the complete homogeneous functions through the Newton recurrence.  Replacing each power sum by $r$ times itself gives \cref{thm:reciprocal_bell,thm:logarithmic_newton_recurrence}.
2362 \end{proof}
2363 
2364 
Statement: Products for the normalized sections · Go to proof · Read statement and proof side by side2365 \begin{proposition}[Products for the normalized sections]\label{prop:normalized_section_product}
2366 For every $0\leq j<N$, the entire function
2367 \[
2368 \Phi_{N,j}(w)
2369 =
2370 j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!}
2371 \]
2372 has order $1/N$.  If $N\geq2$ and $\Lambda_{N,j}$ is its nonzero zero multiset, then
2373 \[
2374 \Phi_{N,j}(w)
2375 =
2376 \prod_{\mu\in\Lambda_{N,j}}
2377 \left(1-\frac w\mu\right).
2378 \]
2379 For $m\geq1$, put
2380 \[
2381 \sigma_{N,j,m}
2382 =
2383 \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}.
2384 \]
2385 Then
2386 \[
2387 C_{N,j,n}(r)
2388 =
2389 \frac{(Nn)!}{n!}
2390 B_n\bigl(
2391  r\sigma_{N,j,1},
2392  r1!\sigma_{N,j,2},
2393  \ldots,
2394  r(n-1)!\sigma_{N,j,n}
2395 \bigr).
2396 \]
2397 \end{proposition}
2398 
Proof block: Products for the normalized sections · Go to statement · Read statement and proof side by side2399 \begin{proof}
2400 The coefficient formula for the order gives $1/N$.  The function is not a polynomial.  An entire function of order below one with only finitely many zeros would be a polynomial times a constant by Hadamard factorization, which is impossible here.  The genus zero product follows as in \cref{thm:genus_zero_product}.
2401 
2402 Taking the logarithm gives
2403 \[
2404 -\log\Phi_{N,j}(w)
2405 =
2406 \sum_{m\geq1}\sigma_{N,j,m}\frac{w^m}{m}.
2407 \]
2408 Since $\Cfun_{N,j}(z)^r=\Phi_{N,j}(z^N)^{-r}$, the Bell polynomial calculation from \cref{thm:reciprocal_bell} gives the coefficient formula.
2409 \end{proof}
2410 
Statement: Definition: def:zero_power_sums2411 \begin{definition}\label{def:zero_power_sums}
2412 For $m\geq1$, define
2413 \[
2414 \sigma_{N,m}
2415 =
2416 \sum_{\lambda\in\Lambda_N}\lambda^{-m},
2417 \]
2418 with multiplicities.
2419 \end{definition}
2420 
2421 The series converges absolutely because the exponent of convergence of $\Lambda_N$ is $1/N<1$.
2422 
Statement: Newton sums for the reduced zeros · Go to proof · Read statement and proof side by side2423 \begin{theorem}[Newton sums for the reduced zeros]\label{thm:newton_sums}
2424 For $|w|$ below the least zero modulus, with the analytic logarithm
2425 normalized to zero at the origin,
2426 \[
2427 -\log\Delta_N(w)
2428 =
2429 \sum_{m\geq1}\sigma_{N,m}\frac{w^m}{m}.
2430 \]
2431 Thus
2432 \[
2433 \kappa_{N,m}=\sigma_{N,m}.
2434 \]
2435 If
2436 \[
2437 a_m=\frac{(-1)^m}{(Nm)!},
2438 \qquad a_0=1,
2439 \]
2440 then
2441 \[
2442 \sigma_{N,m}
2443 =
2444 -ma_m-
2445 \sum_{k=1}^{m-1}a_k\sigma_{N,m-k}.
2446 \]
2447 \end{theorem}
2448 
Proof block: Newton sums for the reduced zeros · Go to statement · Read statement and proof side by side2449 \begin{proof}
2450 Take the logarithm of the product in \cref{thm:genus_zero_product}.  Absolute convergence in a disk below the first zero permits termwise expansion
2451 \[
2452 -\log\left(1-\frac w\lambda\right)
2453 =
2454 \sum_{m\geq1}\frac{w^m}{m\lambda^m}.
2455 \]
2456 This gives the logarithmic formula and identifies $\kappa_{N,m}$ with $\sigma_{N,m}$.
2457 
2458 Differentiate the product identity in logarithmic form to obtain
2459 \[
2460 \Delta_N'(w)
2461 =
2462 -\Delta_N(w)
2463 \sum_{m\geq1}\sigma_{N,m}w^{m-1}.
2464 \]
2465 Comparison of the coefficient of $w^{m-1}$ gives
2466 \[
2467 ma_m
2468 =
2469 -\sum_{k=0}^{m-1}a_k\sigma_{N,m-k}.
2470 \]
2471 Solving for $\sigma_{N,m}$ gives the recurrence.
2472 \end{proof}
2473 
2474 
2475 
Statement: Convergent Mittag Leffler partial-fraction form · Go to proof · Read statement and proof side by side2476 \begin{corollary}[Convergent Mittag--Leffler partial-fraction form]\label{cor:mittag_leffler_tangent}
2477 Away from the zeros of $\Dfun_N$,
2478 \[
2479 \Tfun_N(z)
2480 =
2481 Nz^{N-1}
2482 \sum_{\lambda\in\Lambda_N}
2483 \frac1{\lambda-z^N}.
2484 \]
2485 The series converges absolutely and locally uniformly on compact sets
2486 away from the poles.  Thus no regularizing terms are required.
2487 \end{corollary}
2488 
Proof block: Convergent Mittag Leffler partial-fraction form · Go to statement · Read statement and proof side by side2489 \begin{proof}
2490 Logarithmic differentiation of the genus zero product gives
2491 \[
2492 -\frac{\Delta_N'(w)}{\Delta_N(w)}
2493 =
2494 \sum_{\lambda\in\Lambda_N}\frac1{\lambda-w}.
2495 \]
2496 The series is locally uniform because $\sum_{\lambda}|\lambda|^{-1}<\infty$.  Substitute $w=z^N$ and use the chain rule.
2497 \end{proof}
2498 
Statement: Theta Bell polynomial formula · Go to proof · Read statement and proof side by side2499 \begin{proposition}[Theta Bell polynomial formula]\label{prop:theta_bell}
2500 Put
2501 \[
2502 \widetilde{\sigma}_{N,m}=(-1)^m\sigma_{N,m}.
2503 \]
2504 Then
2505 \[
2506 \Thetafun_{N,n}(u)
2507 =
2508 \frac{(Nn)!}{n!}
2509 B_n\bigl(
2510  \widetilde{\sigma}_{N,1}(u^N+u),
2511  1!\widetilde{\sigma}_{N,2}(u^{2N}+u),
2512  \ldots,
2513  (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u)
2514 \bigr).
2515 \]
2516 \end{proposition}
2517 
Proof block: Theta Bell polynomial formula · Go to statement · Read statement and proof side by side2518 \begin{proof}
2519 The reduced unsigned denominator is
2520 \[
2521 \Psi_N(w)=\sum_{n\geq0}\frac{w^n}{(Nn)!}=\Delta_N(-w).
2522 \]
2523 By \cref{thm:newton_sums},
2524 \[
2525 -\log\Psi_N(w)
2526 =
2527 \sum_{m\geq1}\widetilde{\sigma}_{N,m}\frac{w^m}{m}.
2528 \]
2529 With $w=z^N$, the logarithm of the theta generating function is
2530 \[
2531 \sum_{m\geq1}
2532 \widetilde{\sigma}_{N,m}(u^{Nm}+u)\frac{w^m}{m}.
2533 \]
2534 The complete Bell polynomial formula follows.
2535 \end{proof}
2536 
Statement: Tangent coefficients as zero sums · Go to proof · Read statement and proof side by side2537 \begin{corollary}[Tangent coefficients as zero sums]\label{cor:tangent_zero_sums}
2538 For every $n\geq0$,
2539 \[
2540 T_N(n)
2541 =
2542 N(Nn+N-1)!\sigma_{N,n+1}.
2543 \]
2544 \end{corollary}
2545 
Proof block: Tangent coefficients as zero sums · Go to statement · Read statement and proof side by side2546 \begin{proof}
2547 Combine \cref{prop:tangent_kappa,thm:newton_sums}.
2548 \end{proof}
2549 
Statement: Reciprocal coefficients from zero data · Go to proof · Read statement and proof side by side2550 \begin{corollary}[Reciprocal coefficients from zero data]\label{cor:reciprocal_zero_data}
2551 For every $n\geq0$,
2552 \[
2553 P_{N,n}(r)
2554 =
2555 \frac{(Nn)!}{n!}
2556 B_n\bigl(
2557  r\sigma_{N,1},
2558  r1!\sigma_{N,2},
2559  \ldots,
2560  r(n-1)!\sigma_{N,n}
2561 \bigr).
2562 \]
2563 \end{corollary}
2564 
Proof block: Reciprocal coefficients from zero data · Go to statement · Read statement and proof side by side2565 \begin{proof}
2566 Use \cref{thm:reciprocal_bell,thm:newton_sums}.
2567 \end{proof}
2568 
2569 \begin{example}[The cosine orbit sums]\label{ex:cosine_orbit_sums}
2570 For $N=2$,
2571 \[
2572 \Delta_2(w)=\cos\sqrt w.
2573 \]
2574 Its zeros are
2575 \[
2576 \lambda_m=\pi^2\left(m+\frac12\right)^2,
2577 \qquad m\geq0.
2578 \]
2579 Hence
2580 \[
2581 \sigma_{2,q}
2582 =
2583 \frac{2^{2q}-1}{\pi^{2q}}\zeta(2q).
2584 \]
2585 The identity in \cref{cor:tangent_zero_sums} recovers the classical tangent coefficients from these zero power sums.
2586 \end{example}
2587 
2588 \section{Coefficient asymptotics from the nearest zeros}\label{sec:coefficient_asymptotics}
2589 
2590 For an entire function $f$, let $\Zcal(f)$ denote its multiset of nonzero zeros.  Let $r$ be a positive integer.  Then $\Dfun_N^{-r}$ is meromorphic and a simple zero of $\Dfun_N$ becomes a pole of order $r$.
2591 
Statement: Nearest-pole coefficient expansion · Go to proof · Read statement and proof side by side2592 \begin{theorem}[Nearest-pole coefficient expansion]\label{thm:nearest_circle_expansion}
2593 Let $r\in\ZZ_{>0}$.  Suppose the zeros of $\Dfun_N$ of least modulus
2594 $R$ are simple and form the finite set $\Zcal_R$.  Suppose the next
2595 zero modulus is $R_2>R$.  At each $\rho\in\Zcal_R$, write the principal
2596 part
2597 \[
2598 \Dfun_N(z)^{-r}
2599 =
2600 \sum_{\ell=1}^{r}
2601  a_{\rho,\ell}
2602 \left(1-\frac z\rho\right)^{-\ell}
2603 +
2604 O(1).
2605 \]
2606 Then, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\to\infty$,
2607 \[
2608 P_{N,n}(r)
2609 =
2610  m!
2611 \sum_{\rho\in\Zcal_R}
2612 \sum_{\ell=1}^{r}
2613  a_{\rho,\ell}
2614 \binom{m+\ell-1}{\ell-1}
2615 \rho^{-m}
2616 +
2617 O_{R_*}(m!R_*^{-m}).
2618 \]
2619 The leading Laurent coefficient is
2620 \[
2621 a_{\rho,r}
2622 =
2623 (-\rho\Dfun_N'(\rho))^{-r}.
2624 \]
2625 \end{theorem}
2626 
Proof block: Nearest-pole coefficient expansion · Go to statement · Read statement and proof side by side2627 \begin{proof}
2628 Subtract the complete principal parts at the zeros on $|z|=R$.  The remainder is holomorphic on $|z|<R_2$, hence on a neighborhood of every closed disk $|z|\leq R_*<R_2$.  Cauchy's estimate gives an ordinary coefficient bound $O_{R_*}(R_*^{-m})$.  The coefficient of $z^m$ in
2629 \[
2630 \left(1-\frac z\rho\right)^{-\ell}
2631 \]
2632 is
2633 \[
2634 \binom{m+\ell-1}{\ell-1}\rho^{-m}.
2635 \]
2636 Multiplication by $m!$ converts the ordinary coefficient to $P_{N,n}(r)$.
2637 
2638 Near a simple zero,
2639 \[
2640 \Dfun_N(z)
2641 =
2642 -\rho\Dfun_N'(\rho)
2643 \left(1-\frac z\rho\right)
2644 +O\left(\left(1-\frac z\rho\right)^2\right).
2645 \]
2646 Raising to the power $-r$ gives the formula for $a_{\rho,r}$.
2647 \end{proof}
2648 
Statement: Leading dominant zero term · Go to proof · Read statement and proof side by side2649 \begin{corollary}[Leading dominant zero term]\label{cor:leading_dominant_zero}
2650 Under the hypotheses of \cref{thm:nearest_circle_expansion},
2651 \[
2652 \begin{aligned}
2653 P_{N,n}(r)
2654 ={}&m!\binom{m+r-1}{r-1}
2655 \sum_{\rho\in\Zcal_R}
2656 \frac{\rho^{-m}}{(-\rho\Dfun_N'(\rho))^r}
2657 \\
2658 &+O\left(m!m^{r-2}R^{-m}\right)
2659 +O_{R_*}\left(m!R_*^{-m}\right).
2660 \end{aligned}
2661 \]
2662 For $r=1$, the first error term is absent.
2663 \end{corollary}
2664 
Proof block: Leading dominant zero term · Go to statement · Read statement and proof side by side2665 \begin{proof}
2666 The terms with $\ell<r$ in \cref{thm:nearest_circle_expansion} are bounded by a constant times $m^{r-2}R^{-m}$.  The term with $\ell=r$ gives the displayed main term.
2667 \end{proof}
2668 
2669 
Statement: Orbit reduced nearest circle term · Go to proof · Read statement and proof side by side2670 \begin{corollary}[Orbit reduced nearest circle term]\label{cor:orbit_reduced_asymptotic}
2671 Assume the hypotheses of \cref{thm:nearest_circle_expansion}.  Choose one representative from each orbit of the nearest zeros under multiplication by $\omega$, and call the resulting set $\Zcal_R^{\mathrm{orb}}$.  With $m=Nn$,
2672 \[
2673 \begin{aligned}
2674 P_{N,n}(r)
2675 ={}&N m!\binom{m+r-1}{r-1}
2676 \sum_{\tau\in\Zcal_R^{\mathrm{orb}}}
2677 \frac{\tau^{-m}}{(-\tau\Dfun_N'(\tau))^r}
2678 \\
2679 &+O\left(m!m^{r-2}R^{-m}\right)
2680 +O_{R_*}\left(m!R_*^{-m}\right).
2681 \end{aligned}
2682 \]
2683 For $r=1$, the first error term is absent.
2684 \end{corollary}
2685 
Proof block: Orbit reduced nearest circle term · Go to statement · Read statement and proof side by side2686 \begin{proof}
2687 If $\rho=\omega^k\tau$, then
2688 \[
2689 \Dfun_N'(\rho)=\omega^{-k}\Dfun_N'(\tau).
2690 \]
2691 Therefore
2692 \[
2693 -\rho\Dfun_N'(\rho)
2694 =-\tau\Dfun_N'(\tau).
2695 \]
2696 Since $m=Nn$, one also has $\rho^{-m}=\tau^{-m}$.  Every orbit therefore contributes $N$ equal terms in \cref{cor:leading_dominant_zero}.
2697 \end{proof}
2698 
Statement: Secant asymptotic · Go to proof · Read statement and proof side by side2699 \begin{corollary}[Secant asymptotic]\label{cor:secant_asymptotic}
2700 For $N=2$ and $r=1$,
2701 \[
2702 P_{2,n}(1)
2703 \sim
2704 2(2n)!\left(\frac2\pi\right)^{2n+1}.
2705 \]
2706 \end{corollary}
2707 
Proof block: Secant asymptotic · Go to statement · Read statement and proof side by side2708 \begin{proof}
2709 The nearest zeros of $\cos z$ are $\rho=\pm\pi/2$, both simple.  Their contributions in \cref{cor:leading_dominant_zero} are equal.
2710 \end{proof}
2711 
2712 \section{Zero zeta functions}\label{sec:zero_zeta}
2713 
2714 The assumption $N\geq2$ remains in force throughout this section.  The zero counting theorem gives the exact abscissae of absolute convergence.  The exponentially accurate zero locations give more.  They compare the zero zeta with a Hurwitz zeta function.
2715 
Statement: Definition: def:absolute_zero_zetas2716 \begin{definition}\label{def:absolute_zero_zetas}
2717 Define
2718 \[
2719 \mathfrak Z_N^{\mathrm{abs}}(s)
2720 =
2721 \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
2722 \]
2723 and
2724 \[
2725 \mathfrak Z_N^{\mathrm{red}}(s)
2726 =
2727 \sum_{\lambda\in\Lambda_N}|\lambda|^{-s},
2728 \]
2729 with multiplicities.  Each term means $e^{-s\log|\rho|}$ or $e^{-s\log|\lambda|}$.
2730 \end{definition}
2731 
Statement: Abscissae of absolute convergence · Go to proof · Read statement and proof side by side2732 \begin{theorem}[Abscissae of absolute convergence]\label{thm:zeta_convergence}
2733 The series $\mathfrak Z_N^{\mathrm{abs}}$ converges absolutely and locally uniformly when
2734 \[
2735 \Re s>1.
2736 \]
2737 It fails to converge absolutely when $\Re s\leq1$.  The series $\mathfrak Z_N^{\mathrm{red}}$ converges absolutely and locally uniformly when
2738 \[
2739 \Re s>\frac1N.
2740 \]
2741 It fails to converge absolutely when $\Re s\leq1/N$.  Both functions are holomorphic on their open half planes of absolute convergence.
2742 \end{theorem}
2743 
Proof block: Abscissae of absolute convergence · Go to statement · Read statement and proof side by side2744 \begin{proof}
2745 By \cref{thm:zero_counting}, the zeros of $\Dfun_N$ have linear counting density, while the reduced zeros have counting order $R^{1/N}$.  Partial summation gives absolute convergence in the stated open half planes.  The positive terms obtained after taking absolute values diverge on the boundary and to its left.  Local uniform convergence follows by the Weierstrass test on compact subsets.
2746 \end{proof}
2747 
Statement: Meromorphic continuation of the symmetry-orbit zero zeta · Go to proof · Read statement and proof side by side2748 \begin{theorem}[Meromorphic continuation of the symmetry-orbit zero zeta]\label{thm:zeta_continuation}
2749 Put
2750 \[
2751 A_N=\frac\pi{\sin(\pi/N)}.
2752 \]
2753 There is an entire function $Q_N^{\mathrm{red}}$ such that
2754 \[
2755 \mathfrak Z_N^{\mathrm{red}}(s)
2756 =
2757 A_N^{-Ns}\zeta\left(Ns,\frac12\right)+Q_N^{\mathrm{red}}(s)
2758 \]
2759 on $\Re s>1/N$.  The right hand side gives a meromorphic continuation to $\CC$ with one simple pole at $s=1/N$.  Its residue is
2760 \[
2761 \frac{\sin(\pi/N)}{N\pi}.
2762 \]
2763 
2764 There is an entire function $Q_N^{\mathrm{abs}}$ such that, initially on
2765 $\Re s>1$,
2766 \[
2767 \mathfrak Z_N^{\mathrm{abs}}(s)
2768 =
2769 N A_N^{-s}\zeta\left(s,\frac12\right)+Q_N^{\mathrm{abs}}(s).
2770 \]
2771 Thus $\mathfrak Z_N^{\mathrm{abs}}$ continues meromorphically to $\CC$ with one simple pole at $s=1$, of residue
2772 \[
2773 \frac{N\sin(\pi/N)}\pi.
2774 \]
2775 \end{theorem}
2776 
Proof block: Meromorphic continuation of the symmetry-orbit zero zeta · Go to statement · Read statement and proof side by side2777 \begin{proof}
2778 For $N=2$, \cref{cor:case_two_zeros} and
2779 $\Dfun_2(z)=\Delta_2(z^2)$ give
2780 \[
2781 \lambda_m=\pi^2\left(m+\frac12\right)^2,
2782 \qquad m\geq0.
2783 \]
2784 Thus the first formula holds with $Q_2^{\mathrm{red}}=0$.
2785 
2786 For $N\geq3$, \cref{cor:reduced_zeros} writes all but finitely many reduced zeros as
2787 \[
2788 \lambda_m
2789 =
2790 \left(A_N\left(m+\frac12\right)\right)^N(1+\varepsilon_m)
2791 \]
2792 where $\lambda_m>0$, $1+\varepsilon_m>0$, and
2793 \[
2794 \varepsilon_m
2795 =
2796 O\left(t_m^{-1}e^{-\gamma t_m}\right).
2797 \]
2798 On every compact set of $s$ values,
2799 \[
2800 (1+\varepsilon_m)^{-s}-1
2801 =
2802 O(|\varepsilon_m|)
2803 \]
2804 uniformly for large $m$.  The difference
2805 \[
2806 \lambda_m^{-s}
2807 -
2808 A_N^{-Ns}\left(m+\frac12\right)^{-Ns}
2809 \]
2810 therefore has a normally convergent sum on every compact subset of
2811 $\CC$.  Indeed, the model power has at most polynomial growth in $m$
2812 on a compact set, while $|\varepsilon_m|$ decays exponentially.  The
2813 finitely many exceptional terms $|\lambda|^{-s}$ and the finitely many
2814 omitted model terms are entire functions of $s$.  Their addition to the
2815 normally convergent tail defines $Q_N^{\mathrm{red}}$ and proves the
2816 first formula. Thus that formula is established for every $N\geq2$.
2817 
2818 In either case, the Hurwitz zeta function has a meromorphic continuation with one simple pole at its first argument equal to one and residue one.  Composition with $Ns$ gives the stated pole and residue.
2819 
2820 For every $N\geq2$, each nonzero zero $\lambda$ of $\Delta_N$ gives $N$ zeros of $\Dfun_N$, each of modulus $|\lambda|^{1/N}$.  Hence
2821 \[
2822 \mathfrak Z_N^{\mathrm{abs}}(s)
2823 =
2824 N\mathfrak Z_N^{\mathrm{red}}(s/N).
2825 \]
2826 Substitution gives the second formula with
2827 \[
2828 Q_N^{\mathrm{abs}}(s)=NQ_N^{\mathrm{red}}(s/N),
2829 \]
2830 which is entire.
2831 \end{proof}
2832 
Statement: Exact zeta models in orders two and four · Go to proof · Read statement and proof side by side2833 \begin{corollary}[Exact zeta models in orders two and four]\label{cor:exact_zeta_models}
2834 For $N=2$,
2835 \[
2836 \mathfrak Z_2^{\mathrm{red}}(s)
2837 =
2838 \pi^{-2s}\zeta\left(2s,\frac12\right)
2839 \]
2840 and
2841 \[
2842 \mathfrak Z_2^{\mathrm{abs}}(s)
2843 =
2844 2\pi^{-s}\zeta\left(s,\frac12\right).
2845 \]
2846 For $N=4$, put $A_4=\sqrt2\pi$.  Then
2847 \[
2848 \mathfrak Z_4^{\mathrm{red}}(s)
2849 =
2850 A_4^{-4s}\zeta\left(4s,\frac12\right)
2851 \]
2852 and
2853 \[
2854 \mathfrak Z_4^{\mathrm{abs}}(s)
2855 =
2856 4A_4^{-s}\zeta\left(s,\frac12\right).
2857 \]
2858 In these two cases the entire correction terms in \cref{thm:zeta_continuation} vanish identically.
2859 \end{corollary}
2860 
Proof block: Exact zeta models in orders two and four · Go to statement · Read statement and proof side by side2861 \begin{proof}
2862 For $N=2$, the reduced zeros are
2863 \[
2864 \pi^2\left(m+\frac12\right)^2,
2865 \qquad m\geq0.
2866 \]
2867 For $N=4$, \cref{ex:fourth_order_zeros} gives the reduced zeros
2868 \[
2869 A_4^4\left(m+\frac12\right)^4,
2870 \qquad m\geq0.
2871 \]
2872 Summing their negative powers proves the reduced formulas.  The absolute formulas follow from
2873 \[
2874 \mathfrak Z_N^{\mathrm{abs}}(s)
2875 =
2876 N\mathfrak Z_N^{\mathrm{red}}(s/N).
2877 \]
2878 \end{proof}
2879 
Statement: Definition: def:branch_orbit_zeta2880 \begin{definition}\label{def:branch_orbit_zeta}
2881 Choose a logarithm for each member of the finite exceptional part of $\Lambda_N$.  Use the real logarithm for the eventual positive sequence.  Define
2882 \[
2883 \mathfrak Z_N^{\mathrm{orb}}(s)
2884 =
2885 \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
2886 \]
2887 This series is initially defined on $\Re s>1/N$.
2888 \end{definition}
2889 
Statement: Branch-dependent symmetry-orbit continuation · Go to proof · Read statement and proof side by side2890 \begin{corollary}[Branch-dependent symmetry-orbit continuation]\label{cor:branch_grouped_continuation}
2891 The branch-dependent orbit zeta $\mathfrak Z_N^{\mathrm{orb}}$ continues meromorphically to $\CC$.  It has the same single pole and residue as $\mathfrak Z_N^{\mathrm{red}}$.  Changing the logarithms of the finite exceptional zeros changes the continuation by an entire function.
2892 \end{corollary}
2893 
Proof block: Branch-dependent symmetry-orbit continuation · Go to statement · Read statement and proof side by side2894 \begin{proof}
2895 The tail consists of positive reduced zeros, so it has the same comparison with the Hurwitz zeta.  Only finitely many terms depend on the branch choices.
2896 \end{proof}
2897 
Statement: Definition: def:derivative_weighted_zeta2898 \begin{definition}\label{def:derivative_weighted_zeta}
2899 Let $q$ be a positive integer.  Choose logarithms of the simple zeros of $\Dfun_N$ with arguments in a fixed bounded interval.  Define
2900 \[
2901 \mathfrak Z_{N,q}^{\mathrm{der}}(s)
2902 =
2903 \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}}
2904 \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
2905 \]
2906 \end{definition}
2907 
Statement: Derivative-weighted convergence · Go to proof · Read statement and proof side by side2908 \begin{theorem}[Derivative-weighted convergence]\label{thm:derivative_weighted_convergence}
2909 If $N\geq3$ and $q\geq1$, the series $\mathfrak Z_{N,q}^{\mathrm{der}}(s)$ converges locally uniformly for every $s\in\CC$.  It therefore defines an entire function.
2910 
2911 For $N=2$, the series converges absolutely and locally uniformly on
2912 $\Re s>1$, where it defines a holomorphic function.
2913 \end{theorem}
2914 
Proof block: Derivative-weighted convergence · Go to statement · Read statement and proof side by side2915 \begin{proof}
2916 For $N\geq3$, \cref{thm:large_zero_strings} gives
2917 \[
2918 |\Dfun_N'(\rho)|^{-1}
2919 \leq
2920 C e^{-c_N|\rho|}
2921 \]
2922 for every sufficiently large simple zero.  Let $L$ be a compact subset
2923 of the $s$ plane.  Because the chosen arguments of the zeros lie in a
2924 fixed bounded interval, there are constants $C_L$ and $M_L$ such that
2925 \[
2926 \sup_{s\in L}|e^{-s\Log\rho}|
2927 \leq C_L|\rho|^{M_L}
2928 \]
2929 for every sufficiently large zero.  Hence the absolute value of the
2930 corresponding summand is at most
2931 \[
2932 C_L'|\rho|^{M_L}e^{-q c_N|\rho|}.
2933 \]
2934 The linear zero counting estimate makes this majorant summable.  The
2935 Weierstrass test gives absolute local uniform convergence of the
2936 series of entire summands, so its sum is entire.  The finite exceptional
2937 set causes no difficulty.
2938 
2939 For $N=2$, the derivative of the cosine has modulus one at every zero.  The assertion then follows from the linear zero spacing.
2940 \end{proof}
2941 
2942 \begin{remark}\label{rem:multiple_zeros}
2943 The derivative weighted sum omits the finitely many zeros that might be multiple.  A treatment of those zeros requires the complete principal parts of $\Dfun_N^{-q}$ rather than the single derivative weight.
2944 \end{remark}
2945 
2946 \clearpage\part{Scope and formal evidence}
2947 \section{Scope boundaries and further questions}\label{sec:further_questions}
2948 
2949 The large-zero theorem settles eventual simplicity but does not classify the
2950 finite exceptional set. A remaining question is whether every zero of
2951 $\Dfun_N$ is simple and every reduced zero of $\Delta_N$ is positive. Both
2952 statements hold for $N=2$ and $N=4$. The argument here proves them only outside
2953 a compact set for general $N$.
2954 
2955 The leading correction in \cref{thm:first_zero_correction} may admit iteration
2956 to a full exponential expansion for $\tau_m$. Such an expansion is an open
2957 problem, not a result of this paper. It could refine the entire correction in
2958 the zero-zeta comparison and lead to explicit special values.
2959 
2960 The symmetry-orbit zero zeta has a meromorphic continuation with one pole. No
2961 functional equation is obtained. A branch dependent sum over the $N$ roots of
2962 each reduced zero must specify logarithms on the rays.
2963 
2964 The coefficient families in
2965 \cref{sec:reciprocal_powers,sec:residue_section,sec:theta} have exact formal
2966 parameter-addition laws. Their arithmetic behavior, including odd-prime
2967 congruences, dyadic weighting, and $p$-adic or Amice--Mahler interpolation,
2968 belongs to Paper IX. General bounded-operator semigroup, resolvent, and
2969 Feshbach theory belongs to Paper VI. Paper III owns general nonlinear
2970 operation trees, while abstract completion and occupancy questions belong to
2971 Paper X. None of those theories is inferred from the scalar complex-analytic
2972 results proved here.
2973 
2974 \section{Formalization ledger for version 1.0.0}\label{sec:formalization_ledger}
2975 
2976 The preserved v1.0.0 release was built with Lean 4 v4.34.0-rc2 and Mathlib revision
2977 \leanname{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}
2978 \cite{deMouraUllrich2021,mathlib2020}. Its 151 theorem-map rows resolve to
2979 1164 unique declarations in the namespace \leanname{Komorebi.PaperIV}:
2980 953 theorem declarations and 211 constructions. The recorded namespace audit scanned
2981 2234 declarations, including 1999 theorem declarations. The historical warning fatal
2982 build and supplied kernel audit allowed only
2983 \leanname{propext}, \leanname{Classical.choice}, and
2984 \leanname{Quot.sound} as standard transitive axioms.
2985 
2986 \begingroup
2987 \small
2988 \begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.70\textwidth}@{}}
2989 \toprule
2990 Manuscript records & Compiled Paper IV scope \\
2991 \midrule
2992 \endfirsthead
2993 \toprule
2994 Manuscript records & Compiled Paper IV scope (continued) \\
2995 \midrule
2996 \endhead
2997 3.1--3.3 & Actual holomorphic root filters, convergent coefficient
2998 selection, orthogonal resolution, the exact range--covariance characterization,
2999 and the complete pointwise cyclic product law. The formal versions hold as well. \\
3000 4.1--4.5 and 5.1--5.8 & Entire sections, finite exponential formulas,
3001 derivative cycles, both Wronskians, and unique entire ODE basis expansions.
3002 Both pointwise addition laws and denominator symmetries are checked.
3003 The exact ordinary hypergeometric formula follows from the omitted-parameter
3004 list, Gamma normalization, and Mathlib's everywhere convergent series. \\
3005 6.1--6.6 & Quotient-algebra power bases and actual exponential-series
3006 convergence in the finite-dimensional coefficient topology. Both pointwise
3007 section-coordinate identities. Signed-shift powers, all matrix-exponential
3008 entries, group laws and trace are checked. The characteristic polynomial
3009 $X^N+1$, its distinct roots, exact spectrum and determinant one are proved. \\
3010 7.1--7.8 & Named reciprocal families, exact parameter recurrences, Bell
3011 formula, parameter degree and leading coefficient, and polynomial divisibility.
3012 Parameter addition holds with any single fixed holomorphic logarithm on the
3013 stated domain. Newton's recurrence gives the literal Hessenberg determinant.
3014 All three compact reciprocal-polynomial formulas are checked. \\
3015 8.1--8.5 & The full tangent coefficient series has the exact kappa and
3016 factorial normalization, including $N=1$. Reciprocal differentiation holds on
3017 every stated fixed logarithm domain. The actual higher Riccati identity holds
3018 at every point where the denominator is nonzero. \\
3019 9.1--9.10 & Arbitrary-parameter residue reciprocals and their analytic germs,
3020 polynomial degree and leading term, sparse Appell laws, joint convolution,
3021 monicity and covariance. The Hasse and derivative-truncation encodings are
3022 proved equal. The named Appell polynomial satisfies both the forward action
3023 and the exact $\Phi_{N,j}(\partial_x^N)^r$ inverse formula. \\
3024 10.1--10.4 & Exact formal theta reconstruction, polynomial lift and degree,
3025 literal divisibility by $u$ and, for even $N$, by $u+1$, and the theta Newton
3026 recurrence with its first coefficient. \\
3027 11.2, 11.4--11.11 & Two-sided maximum-modulus bounds prove exact order
3028 and type. The indicator limsup formula holds in every direction, including ties.
3029 Dominant-exponential errors and zero freeness are uniform on compact sets
3030 avoiding ties. Near cancellation and the order-two zero set are checked.
3031 For $N\geq3$, gap and remainder bounds, sector confinement, global exhaustion
3032 and multiplicity identities give the full large-zero-string theorem.
3033 The real leading correction, quadratic exponential error, signed derivative
3034 asymptotic, eventual alternating sign and cubic correction are checked. \\
3035 11.13--11.15 & The reduced series is entire, and
3036 $\Dfun_N(z)=\Delta_N(z^N)$ preserves multiplicity away from zero. Finite
3037 exceptional sets, eventual positive simple reduced zeros, and their stated
3038 displacement are checked, including $N=2$. Both full disk-counting laws
3039 retain all exceptional multiplicities. The full-zero reciprocal sums have
3040 exact thresholds $s>1$ and $s>1/N$, including boundary divergence. \\
3041 12.1 & For every $N\geq2$, the complete multiplicity-indexed product
3042 converges uniformly on compact sets and equals $\Delta_N$. Jensen and Poisson
3043 estimates prove rigidity of the actual quotient from derived growth bounds.
3044 No growth certificate remains. The original-variable product for $\Dfun_N$
3045 is also checked. \\
3046 12.2--12.4 & The actual diagonal operator is trace class, with a
3047 basis-independent trace and determinant given by limits of actual finite
3048 compressions. Infinite complete homogeneous coefficients are limits of finite
3049 monomial sums. Every normalized section has exact order $1/N$. For $N\geq2$
3050 its full zero product and zero-moment Bell formulas are proved. The actual
3051 complex-power germ at zero has the named residue Taylor coefficients. \\
3052 12.6--12.10 & The normalized logarithm on each zero free disk has the
3053 convergent zero-moment series. Its coefficients equal $\kappa_{N,m}$ and
3054 satisfy Newton's recurrence. The Mittag--Leffler tangent expansion converges
3055 absolutely and locally uniformly away from poles. Tangent coefficients and
3056 the named reciprocal and theta families have their exact zero-moment Bell
3057 formulas. The full tangent series is identified with the analytic function. \\
3058 13.1--13.4 & Analytic division constructs each simple-zero principal part.
3059 Its leading coefficient is uniquely determined. Removing the principal parts
3060 gives a holomorphic remainder and the exact factorial-scaled Cauchy estimate.
3061 Both leading-term errors are bounded, with the first absent for $r=1$.
3062 Rotation gives the orbit-reduced main term. The secant formula holds as a
3063 ratio limit, with an exponentially vanishing relative error. \\
3064 14.2--14.4, 14.6 & The exact all-zero sums retain analytic multiplicity.
3065 Their sharp absolute-convergence domains, boundary divergence and local
3066 uniform convergence are checked. Both Hurwitz comparisons, entire corrections,
3067 global meromorphic continuations, unique simple poles and exact residues are
3068 proved. The order-two/four models have identically zero corrections. Branch
3069 choices on the finite exceptional part change the continuation by an entire
3070 function and preserve its pole and residue. \\
3071 14.8 & The derivative-weighted series uses every actual simple zero and genuine
3072 logarithms with bounded arguments. Absolute and local uniform convergence and
3073 an entire sum are proved for $N\geq3$. For $N=2$, absolute/local uniform
3074 convergence and holomorphy hold on $\Re s>1$. \\
3075 \bottomrule
3076 \end{longtable}
3077 \endgroup
3078 
3079 The declaration-level theorem map is authoritative for the preserved
3080 v1.0.0 contract. Its release records full scoped coverage of all 68
3081 theorem like records: 68 full, zero partial, and zero unmapped. Rows 143--151 close the final fifteen partial records
3082 without altering their statements or assumptions. Imported standard fact
3083 11.1 remains imported. Formal identities, normalized analytic germs, and
3084 fixed logarithm domain identities retain their respective stated meanings.
3085 Finite computations do not change these proof boundaries.
3086 
3087 \section{Conclusion}\label{sec:conclusion}
3088 
3089 Cyclic projection turns the exponential into a finite differential basis, and
3090 the signed cycle produces the distinguished denominator $\Dfun_N$. Its
3091 reciprocal powers yield Bell and Newton recurrences, an additive parameter
3092 law, Appell systems, and theta polynomials. Reduction by $w=z^N$ then connects
3093 the reciprocal coefficients to a genus zero product and reduced-zero power
3094 sums.
3095 
3096 Classical two term localization, specialized to the regular exponent polygon
3097 and combined with reflection symmetry, places all sufficiently large zeros
3098 on symmetry rays. The displacement, derivative and counting estimates yield
3099 the family-specific Hurwitz comparison for the orbit zero zeta.
3100 
3101 \begingroup
3102 \small
3103 \let\prscbibliography\thebibliography
3104 \renewcommand{\thebibliography}[1]{\prscbibliography{#1}\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
3105 \bibliographystyle{alpha}
3106 \bibliography{references}
3107 \endgroup
3108 
3109 \clearpage
3110 \appendix
3111 \section{Visual atlas}\label{app:visual-atlas}
3112 The twenty-six plates retain the worked models of the separate visual guide. Each stable label V1 through V26 names a mathematical mechanism and points back to the relevant statements. The plates distinguish exact identities, rational certificates, sampled curves, and inherited formal evidence.
3113 \begin{readerroadmap}[title={Three routes through the atlas}]
3114 For algebra, read V3 through V16. For zero geometry and its consequences, read V17 through V24. V25 follows one coefficient through the complete calculation. V26 explains the evidence attached to the models.
3115 \end{readerroadmap}
3116 \clearpage
3117 \begingroup
3118 \fontsize{10}{12}\selectfont\setlength{\parskip}{4pt}
3119 \setlength{\abovedisplayskip}{6pt}\setlength{\belowdisplayskip}{6pt}
3120 \input{atlas-pages}
3121 \endgroup
3122 
3123 \end{document}

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