Realization Calculus

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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{}
132 
133 \hypersetup{
134   pdftitle={Projector-Resolved Seed Calculus IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data},
135   pdfauthor={Komorebi P.},
136   pdfsubject={Cyclic exponential sections, reciprocal families, zero geometry, and spectral zero data},
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179 {\sffamily\fontsize{8.5}{11}\selectfont\color{pubTeal}REALIZATION CALCULUS\par}%
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195 \begin{document}
196 \KomorebiCover
197 \KomorebiOpeningNote{Begin with cyclic projection and the signed differential cycle. The reciprocal sections 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.
198 
199 The visual atlas at the end of this chapter supplies twenty-six worked plates. The companion Lean guide follows the formal declarations and their hypotheses. The main text contains sixty-eight theorem-like results and ninety-five numbered records.}
200 
201 \begin{abstract}
202 Fix a positive integer $N$. Root of unity projection of the exponential
203 produces residue-class sections that, for $\lambda\ne0$, form a basis for
204 $y^{(N)}=\lambda^N y$. The signed sections for $y^{(N)}=-y$ yield the
205 denominator $\Dfun_N$ and its reduced form $\Delta_N$. We derive reciprocal
206 coefficient polynomials, Bell and Newton recurrences, an additive parameter
207 law, and residue section Appell and theta families. For $N\geq2$,
208 $\Delta_N$ has order $1/N$. We specialize classical two term zero theory for
209 exponential sums to the regular exponent polygon. In this equal-weight family,
210 conjugation symmetry combined with unique near-model localization places all
211 sufficiently large zeros exactly on the symmetry rays, and direct expansion
212 gives their leading displacement and derivative sign. The classical perimeter
213 law yields zero-counting asymptotics with explicit leading constants and
214 bounded error. These results give genus-zero products, the standard diagonal
215 Fredholm representation, zero-power-sum formulas for reciprocal and
216 logarithmic coefficients, nearest-zero coefficient asymptotics, and
217 meromorphic continuation of the symmetry-orbit zero zeta by comparison with
218 the Hurwitz zeta function. Complex powers are interpreted either formally or
219 through a fixed logarithm on a zero-free domain. This chapter treats only this
220 cyclic analytic family. Arithmetic interpolation, general nonlinear trees,
221 general resolvent theory, and abstract completion are reserved for Chapters IX,
222 III, VI, and X.
223 \end{abstract}
224 
225 \begin{readerroadmap}[title={Formal scope and evidence}]
226 The recorded formal contract covers all sixty-eight theorem-like manuscript results under their stated hypotheses. The formalization ledger identifies the corresponding declarations.
227 
228 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. Finite computations support the worked models. The companion Lean guide records the replay using pinned sources, declaration-kind checks, axiom audit, and their limits.
229 \end{readerroadmap}
230 
231 \clearpage
232 \tableofcontents
233 \listoffigures
234 \clearpage\pagenumbering{arabic}
235 
236 \section{Introduction}\label{sec:introduction}
237 
238 For a power series
239 \[
240 F(z)=\sum_{m\geq 0}a_mz^m
241 \]
242 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}.
243 
244 Chapters I and II developed typed seed grammars, exact projector packets, and
245 finite residue-projector algebra, while Chapter III treated typed edge operators
246 and exact coordinate transport \cite{PaperI2026,PaperII2026,PaperIII2026}.
247 The root filter used below is the finite cyclic scalar specialization of that
248 earlier projector calculus. The signed companion matrix is likewise a concrete
249 cyclic instance, not a replacement for Chapter III's general typed-edge theory.
250 
251 This chapter 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
252 \[
253 y^{(N)}=-y.
254 \]
255 The section with initial value one is
256 \[
257 \Dfun_N(z)=\sum_{n\geq 0}(-1)^n\frac{z^{Nn}}{(Nn)!}.
258 \]
259 Its reciprocal powers produce polynomial coefficient families. Its logarithmic
260 derivative produces a scalar higher Riccati identity. The same reciprocal
261 construction applied to every exponential section gives normalized
262 residue section functions and ordinary Appell polynomial sequences. These
263 cyclic exponential sections and their matrix addition laws belong to the
264 generalized hyperbolic function literature
265 \cite{Schwaiger1992,Muldoon2005}. Appell and binomial-type sequences are
266 discussed in \cite{Roman1984,Comtet1974}.
267 
268 The second half of this chapter concerns the zeros of $\Dfun_N$.  The finite exponential sum
269 \[
270 \Dfun_N(z)=\frac1N\sum_{\eta^N=-1}e^{\eta z}
271 \]
272 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.
273 
274 The rotational symmetry is more useful after the change of variable $w=z^N$.  There is an entire function
275 \[
276 \Delta_N(w)=\sum_{n\geq 0}(-1)^n\frac{w^n}{(Nn)!}
277 \]
278 with
279 \[
280 \Dfun_N(z)=\Delta_N(z^N).
281 \]
282 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}.
283 
284 The cyclic exponential sections and their matrix addition laws have a
285 classical literature. The tools used below include Appell and binomial-type
286 sequences, canonical products, singularity analysis, and Fredholm
287 determinants. These tools are also standard. Our aim is to organize this symmetric
288 specialization and link its reciprocal coefficients to reduced-zero data. No
289 historical priority is asserted for the component constructions.
290 
291 
292 
293 \section{A map of the construction}\label{sec:map}
294 
295 \begin{keyidea}
296 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.
297 \end{keyidea}
298 
299 Two variables appear throughout this chapter.  The variable $z$ carries the differential equation and the rotational symmetry.  The reduced variable
300 \[
301 w=z^N
302 \]
303 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 $w$.
304 
305 The analytic objects are introduced in the following order.  First,
306 \[
307 \Efun_{N,j}(z)
308 =
309 \sum_{n\geq0}\frac{z^{Nn+j}}{(Nn+j)!}
310 \]
311 is the residue $j$ part of the exponential.  Second,
312 \[
313 \Dfun_{N,j}(z)
314 =
315 \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}
316 \]
317 is the signed basis for $y^{(N)}=-y$.  The section $\Dfun_N=\Dfun_{N,0}$ is the denominator used in the reciprocal family
318 \[
319 \Sfun_N^{(r)}=\Dfun_N^{-r}.
320 \]
321 Set $\Tfun_N=-\Dfun_N^{\prime}/\Dfun_N$. Equivalently, $\Tfun_N$ is the logarithmic derivative of $\Sfun_N^{(1)}$.
322 
323 A different reciprocal family comes from normalizing each exponential section at the origin.  The function
324 \[
325 \Cfun_{N,j}(z)
326 =
327 \frac{z^j/j!}{\Efun_{N,j}(z)}
328 \]
329 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.
330 
331 The zero theory begins only after these algebraic relations have been established.  The factorization
332 \[
333 \Dfun_N(z)=\Delta_N(z^N)
334 \]
335 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
336 \[
337 \text{zero power sums}
338 =
339 \text{logarithmic coefficients}.
340 \]
341 This equality is the bridge between the first and second halves of this chapter.
342 
343 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.
344 
345 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.
346 
347 This chapter 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.
348 
349 \clearpage\part{Cyclic resolution and differential transport}
350 \section{Cyclic projection of analytic functions}\label{sec:cyclic_projection}
351 
352 \begin{readerroadmap}
353 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.
354 \end{readerroadmap}
355 \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}
356 
357 
358 Fix $N\geq1$ and write
359 \[
360 \omega=e^{2\pi i/N}.
361 \]
362 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
363 \[
364 (\Bcal_{N,j}F)(z)
365 =
366 \frac1N\sum_{\ell=0}^{N-1}\omega^{-j\ell}F(\omega^\ell z),
367 \qquad 0\leq j<N.
368 \]
369 The same formula is valid for formal power series.
370 
371 \begin{theorem}[Root filter]\label{thm:root_filter}
372 If
373 \[
374 F(z)=\sum_{m\geq0}a_mz^m,
375 \]
376 then
377 \[
378 \Bcal_{N,j}F(z)
379 =
380 \sum_{m\equiv j\bmod N}a_mz^m.
381 \]
382 The operators satisfy
383 \[
384 \Bcal_{N,i}\Bcal_{N,j}=\delta_{ij}\Bcal_{N,j}
385 \]
386 and
387 \[
388 \sum_{j=0}^{N-1}\Bcal_{N,j}=I.
389 \]
390 \end{theorem}
391 
392 \begin{proof}
393 Substitution of the power series gives
394 \[
395 \Bcal_{N,j}F(z)
396 =
397 \sum_{m\geq0}a_mz^m
398 \left(
399 \frac1N\sum_{\ell=0}^{N-1}\omega^{(m-j)\ell}
400 \right).
401 \]
402 The inner average is one when $m\equiv j\bmod N$ and zero otherwise.  The operator identities follow from the disjoint residue classes.
403 \end{proof}
404 
405 \begin{proposition}[Covariance characterization]\label{prop:covariance}
406 A holomorphic function $F$ belongs to the range of $\Bcal_{N,j}$ if and only if
407 \[
408 F(\omega z)=\omega^jF(z).
409 \]
410 \end{proposition}
411 
412 \begin{proof}
413 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
414 \[
415 \sum_k\omega^k\Bcal_{N,k}F
416 =
417 \omega^j\sum_k\Bcal_{N,k}F.
418 \]
419 Application of $\Bcal_{N,k}$ gives
420 \[
421 (\omega^k-\omega^j)\Bcal_{N,k}F=0.
422 \]
423 The powers $1,\omega,\ldots,\omega^{N-1}$ are distinct, so
424 $\Bcal_{N,k}F=0$ for $k\ne j$.
425 \end{proof}
426 
427 Multiplication is not diagonal in these coordinates. The residue labels add
428 modulo $N$. This is the scalar cyclic instance of the packet product from
429 Chapter I.
430 
431 \begin{theorem}[Cyclic product law]\label{thm:cyclic_product}
432 If $F$ and $G$ are holomorphic on the same rotation-invariant domain, then
433 \[
434 \Bcal_{N,k}(FG)
435 =
436 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
437 (\Bcal_{N,i}F)(\Bcal_{N,j}G).
438 \]
439 \end{theorem}
440 
441 \begin{proof}
442 Write $F=\sum_iF_i$ and $G=\sum_jG_j$, where
443 $F_i=\Bcal_{N,i}F$ and $G_j=\Bcal_{N,j}G$.  Their covariance gives
444 \[
445 (F_iG_j)(\omega z)=\omega^{i+j}F_i(z)G_j(z),
446 \]
447 so $F_iG_j$ belongs to the packet labeled by $i+j$ modulo $N$.
448 Expanding the finite product and applying $\Bcal_{N,k}$ gives the formula.
449 \end{proof}
450 
451 \begin{remark}\label{rem:formal_analytic}
452 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.
453 \end{remark}
454 
455 \section{The transfer exponential basis}\label{sec:transfer_exponential}
456 
457 \begin{definition}\label{def:transfer_exponential}
458 For $\lambda\in\CC$ and $0\leq j<N$, define
459 \[
460 \Efun_{N,j}(\lambda z)
461 =
462 \Bcal_{N,j}(e^{\lambda z})
463 =
464 \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
465 \]
466 \end{definition}
467 
468 \begin{theorem}[Differential cycle and initial data]\label{thm:exponential_cycle}
469 For $1\leq j<N$,
470 \[
471 \frac{d}{dz}\Efun_{N,j}(\lambda z)
472 =
473 \lambda\Efun_{N,j-1}(\lambda z),
474 \]
475 while
476 \[
477 \frac{d}{dz}\Efun_{N,0}(\lambda z)
478 =
479 \lambda\Efun_{N,N-1}(\lambda z).
480 \]
481 Thus
482 \[
483 \frac{d^N}{dz^N}\Efun_{N,j}(\lambda z)
484 =
485 \lambda^N\Efun_{N,j}(\lambda z).
486 \]
487 For $0\leq k<N$,
488 \[
489 \left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0}
490 =
491 \delta_{kj}\lambda^j.
492 \]
493 If $\lambda\ne0$, the $N$ functions $\Efun_{N,j}(\lambda z)$ form a basis of the solution space of $y^{(N)}=\lambda^Ny$.
494 \end{theorem}
495 
496 \begin{proof}
497 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.
498 \end{proof}
499 
500 
501 \begin{corollary}[Wronskian of the transfer basis]\label{cor:wronskians}
502 Assume $N\geq2$. For $\lambda\ne0$, the Wronskian of the ordered basis
503 \[
504 \Efun_{N,0}(\lambda z),\ldots,\Efun_{N,N-1}(\lambda z)
505 \]
506 is the constant
507 \[
508 \lambda^{N(N-1)/2}.
509 \]
510 \end{corollary}
511 
512 \begin{proof}
513 At the origin, the derivative matrix is diagonal with entries
514 \[
515 1,\lambda,\ldots,\lambda^{N-1}.
516 \]
517 For an explicit determinant proof, write each basis function as a finite
518 sum of exponentials with exponents $a_\ell=\lambda\omega^\ell$.
519 Its derivative matrix factors as
520 \[
521 W(z)=V\,\operatorname{diag}(e^{a_\ell z})\,C,
522 \qquad V_{k\ell}=a_\ell^k,
523 \]
524 where $C$ is the constant matrix of Fourier weights. Thus
525 \[
526 \det W(z)=e^{z\sum_\ell a_\ell}\det W(0)=\det W(0),
527 \]
528 because $N\geq2$ implies $\sum_\ell\omega^\ell=0$.
529 The product of the diagonal initial values is the stated power of $\lambda$.
530 \end{proof}
531 
532 \begin{corollary}[Addition law]\label{cor:exponential_addition}
533 For $0\leq k<N$,
534 \[
535 \Efun_{N,k}(\lambda(z+w))
536 =
537 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
538 \Efun_{N,i}(\lambda z)\Efun_{N,j}(\lambda w).
539 \]
540 \end{corollary}
541 
542 \begin{proof}
543 Expand $e^{\lambda(z+w)}=e^{\lambda z}e^{\lambda w}$ as a
544 bivariate power series.  A term $z^a w^b$ contributes to the
545 $k$th section exactly when $a+b\equiv k\bmod N$.  Grouping first by
546 $i\equiv a\bmod N$ and $j\equiv b\bmod N$ gives the displayed sum.
547 \end{proof}
548 
549 The basis also has a scalar hypergeometric form.
550 
551 \begin{theorem}[Hypergeometric form]\label{thm:hypergeometric_form}
552 Let
553 \[
554 A_{N,j}
555 =
556 \left\{
557 \frac{j+m}{N}
558 \mathrel{:}
559 1\leq m\leq N
560 \text{ and }
561 j+m\ne N
562 \right\}.
563 \]
564 Then
565 \[
566 \Efun_{N,j}(\lambda z)
567 =
568 \frac{(\lambda z)^j}{j!}
569 {}_0F_{N-1}
570 \left(
571 \begin{matrix}-\\ A_{N,j}\end{matrix}
572 \mathrel{,}
573 \frac{(\lambda z)^N}{N^N}
574 \right).
575 \]
576 \end{theorem}
577 
578 \begin{proof}
579 The multiplication formula for the gamma function gives
580 \[
581 (Nn+j)!
582 =
583 j!N^{Nn}n!
584 \prod_{a\in A_{N,j}}(a)_n.
585 \]
586 Indeed, the $N$ Pochhammer factors supplied by the multiplication formula
587 are $((j+m)/N)_n$ for $1\leq m\leq N$. The excluded factor is
588 $((j+N-j)/N)_n=(1)_n=n!$.
589 Insertion into the defining series gives the stated hypergeometric expansion.
590 \end{proof}
591 
592 \begin{example}\label{ex:classical_exponential}
593 For $N=1$, the only section is $e^{\lambda z}$.  For $N=2$,
594 \[
595 \Efun_{2,0}(z)=\cosh z,
596 \qquad
597 \Efun_{2,1}(z)=\sinh z.
598 \]
599 These are classical instances of the same cyclic basis.
600 \end{example}
601 
602 \section{Signed cyclic sections and the distinguished denominator}\label{sec:signed_system}
603 
604 \begin{definition}\label{def:signed_sections}
605 For $0\leq j<N$, define
606 \[
607 \Dfun_{N,j}(z)
608 =
609 \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}.
610 \]
611 The distinguished signed denominator is
612 \[
613 \Dfun_N(z)=\Dfun_{N,0}(z).
614 \]
615 \end{definition}
616 
617 Fix any $\rho\in\CC$ with $\rho^N=-1$.
618 
619 \begin{proposition}[Relation with the transfer exponential]\label{prop:signed_exponential_relation}
620 For every $j$,
621 \[
622 \Dfun_{N,j}(z)
623 =
624 \rho^{-j}\Efun_{N,j}(\rho z).
625 \]
626 \end{proposition}
627 
628 \begin{proof}
629 The coefficient of $z^{Nn+j}$ on the right is
630 \[
631 \rho^{-j}\rho^{Nn+j}\frac1{(Nn+j)!}
632 =
633 (-1)^n\frac1{(Nn+j)!}.
634 \]
635 \end{proof}
636 
637 \begin{theorem}[Signed derivative cycle]\label{thm:signed_cycle}
638 For $1\leq j<N$,
639 \[
640 \Dfun_{N,j}'=\Dfun_{N,j-1},
641 \]
642 and
643 \[
644 \Dfun_{N,0}'=-\Dfun_{N,N-1}.
645 \]
646 Every signed section satisfies
647 \[
648 \Dfun_{N,j}^{(N)}=-\Dfun_{N,j}.
649 \]
650 The $N$ signed sections form the basis of $y^{(N)}=-y$ determined by
651 \[
652 \Dfun_{N,j}^{(k)}(0)=\delta_{jk}
653 \]
654 for $0\leq j,k<N$.
655 \end{theorem}
656 
657 \begin{proof}
658 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.
659 \end{proof}
660 
661 \begin{corollary}[Wronskian of the signed basis]\label{cor:signed_wronskian}
662 Assume $N\geq2$. The Wronskian of
663 \[
664 \Dfun_{N,0}(z),\ldots,\Dfun_{N,N-1}(z)
665 \]
666 is one.
667 \end{corollary}
668 
669 \begin{proof}
670 The derivative matrix at the origin is the identity. The determinant
671 factorization in the proof of \cref{cor:wronskians} applies with
672 $a_\ell=\rho\omega^\ell$ and with the signed-section Fourier weights.
673 Since $\sum_\ell a_\ell=0$, the determinant is constant and equals one.
674 \end{proof}
675 
676 
677 \begin{proposition}[Twisted addition law]\label{prop:signed_addition}
678 For $0\leq k<N$,
679 \[
680 \Dfun_{N,k}(z+w)
681 =
682 \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}
683 (-1)^{\lfloor(i+j)/N\rfloor}
684 \Dfun_{N,i}(z)\Dfun_{N,j}(w).
685 \]
686 \end{proposition}
687 
688 \begin{proof}
689 Use \cref{prop:signed_exponential_relation,cor:exponential_addition}.  The factor introduced by a pair $(i,j)$ is
690 \[
691 \rho^{i+j-k}.
692 \]
693 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 $i+j=k+N$.
694 \end{proof}
695 
696 \begin{theorem}[Finite exponential representation]\label{thm:finite_exponential_representation}
697 Let
698 \[
699 \Rcal_N=\{\eta\in\CC\mathrel{:}\eta^N=-1\}.
700 \]
701 Then
702 \[
703 \Dfun_N(z)
704 =
705 \frac1N\sum_{\eta\in\Rcal_N}e^{\eta z}.
706 \]
707 More generally,
708 \[
709 \Dfun_{N,j}(z)
710 =
711 \frac{\rho^{-j}}N
712 \sum_{\ell=0}^{N-1}
713 \omega^{-j\ell}e^{\rho\omega^\ell z}.
714 \]
715 \end{theorem}
716 
717 \begin{proof}
718 The second formula is the root filter expression for $\rho^{-j}\Efun_{N,j}(\rho z)$.  The first is the case $j=0$.
719 \end{proof}
720 
721 \begin{definition}\label{def:reduced_denominator}
722 Define the reduced denominator
723 \[
724 \Delta_N(w)
725 =
726 \sum_{n\geq0}(-1)^n\frac{w^n}{(Nn)!}.
727 \]
728 Then
729 \[
730 \Dfun_N(z)=\Delta_N(z^N).
731 \]
732 \end{definition}
733 
734 \begin{corollary}[Symmetries]\label{cor:denominator_symmetries}
735 The denominator satisfies
736 \[
737 \Dfun_N(\omega z)=\Dfun_N(z)
738 \]
739 and
740 \[
741 \Dfun_N(\overline z)=\overline{\Dfun_N(z)}.
742 \]
743 Its nonzero zeros occur in orbits under multiplication by the $N$th roots of unity and under complex conjugation.
744 \end{corollary}
745 
746 \begin{proof}
747 The first identity follows from the powers $z^{Nn}$.  The second follows from the real coefficients.  Both symmetries preserve the zero set and multiplicity.
748 \end{proof}
749 
750 \begin{example}\label{ex:small_denominators}
751 The first denominators are
752 \[
753 \Dfun_1(z)=e^{-z},
754 \qquad
755 \Dfun_2(z)=\cos z.
756 \]
757 For $N=3$,
758 \[
759 \Dfun_3(z)
760 =
761 \frac13\left(
762 e^{-z}
763 +2e^{z/2}\cos\left(\frac{\sqrt3z}{2}\right)
764 \right).
765 \]
766 For $N=4$,
767 \[
768 \Dfun_4(z)
769 =
770 \cos\left(\frac z{\sqrt2}\right)
771 \cosh\left(\frac z{\sqrt2}\right).
772 \]
773 \end{example}
774 
775 \section{A finite algebra and matrix model}\label{sec:finite_algebra}
776 
777 \begin{keyidea}
778 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.
779 \end{keyidea}
780 \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}
781 
782 
783 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.
784 
785 \begin{definition}\label{def:cyclic_algebras}
786 Let
787 \[
788 \mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
789 \]
790 and
791 \[
792 \mathfrak A_N^-=\CC[\xi]/(\xi^N+1).
793 \]
794 Every element has a unique representative of degree below $N$.
795 \end{definition}
796 
797 \begin{theorem}[Finite algebra exponential identities]\label{thm:finite_algebra_exponentials}
798 In $\mathfrak A_N^+$ one has
799 \[
800 e^{\lambda z\xi}
801 =
802 \sum_{j=0}^{N-1}\Efun_{N,j}(\lambda z)\xi^j.
803 \]
804 In $\mathfrak A_N^-$ one has
805 \[
806 e^{z\xi}
807 =
808 \sum_{j=0}^{N-1}\Dfun_{N,j}(z)\xi^j.
809 \]
810 Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.
811 \end{theorem}
812 
813 \begin{proof}
814 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.
815 
816 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
817 \[
818 e^{(z+w)\xi}=e^{z\xi}e^{w\xi}
819 \]
820 gives the addition laws after comparison of the coefficients of $1,\xi,\ldots,\xi^{N-1}$.
821 \end{proof}
822 
823 \begin{definition}\label{def:signed_shift_matrix}
824 Let $J_N$ be the signed cyclic shift on the standard basis $e_0,\ldots,e_{N-1}$ of $\CC^N$, defined by
825 \[
826 J_Ne_j=e_{j+1}
827 \quad (0\leq j<N-1),
828 \qquad
829 J_Ne_{N-1}=-e_0.
830 \]
831 Then $J_N^N=-I$.
832 \end{definition}
833 
834 \begin{theorem}[Companion matrix realization]\label{thm:companion_matrix_realization}
835 Put
836 \[
837 \mathbf D_N(z)
838 =
839 \begin{pmatrix}
840 \Dfun_{N,0}(z)\\
841 \Dfun_{N,1}(z)\\
842 \vdots\\
843 \Dfun_{N,N-1}(z)
844 \end{pmatrix}.
845 \]
846 Then
847 \[
848 \mathbf D_N(z)=e^{zJ_N}e_0
849 \]
850 and
851 \[
852 \Dfun_N(z)=\frac1N\tr(e^{zJ_N}).
853 \]
854 The eigenvalues of $J_N$ are the $N$ roots of $-1$, each with multiplicity one.
855 \end{theorem}
856 
857 \begin{proof}
858 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.
859 
860 The vectors
861 \[
862 e_0,J_Ne_0,\ldots,J_N^{N-1}e_0
863 \]
864 are the standard basis.  Thus $e_0$ is cyclic for $J_N$, so the
865 minimal polynomial of $J_N$ has degree $N$.  Since $J_N^N=-I$, that
866 minimal polynomial divides $x^N+1$ and therefore equals $x^N+1$.
867 The characteristic polynomial has the same degree and is divisible
868 by the minimal polynomial, so it too equals $x^N+1$.  This polynomial
869 has distinct roots.  Consequently $J_N$ is diagonalizable and its
870 spectrum consists of all $N$ roots of $-1$, each once.  Hence
871 \[
872 \tr(e^{zJ_N})
873 =
874 \sum_{\eta^N=-1}e^{\eta z}
875 =
876 N\Dfun_N(z)
877 \]
878 by \cref{thm:finite_exponential_representation}.
879 \end{proof}
880 
881 \begin{corollary}[Entries of the fundamental matrix]\label{cor:fundamental_matrix_entries}
882 For $0\leq p,q<N$, let $d_{pq}$ be the member of $\{0,\ldots,N-1\}$ congruent to $p-q$ modulo $N$.  Put
883 \[
884 \varepsilon_{pq}
885 =
886 \begin{cases}
887 1,&p\geq q,\\
888 -1,&p<q.
889 \end{cases}
890 \]
891 Then
892 \[
893 \bigl(e^{zJ_N}\bigr)_{pq}
894 =
895 \varepsilon_{pq}\Dfun_{N,d_{pq}}(z).
896 \]
897 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.
898 \end{corollary}
899 
900 \begin{proof}
901 For every nonnegative integer $n$,
902 \[
903 J_N^{Nn+d_{pq}}e_q
904 =
905 (-1)^n\varepsilon_{pq}e_p.
906 \]
907 Only powers whose exponent is congruent to $d_{pq}$ modulo $N$ contribute to the $(p,q)$ entry of the exponential.  Therefore
908 \[
909 \bigl(e^{zJ_N}\bigr)_{pq}
910 =
911 \varepsilon_{pq}
912 \sum_{n\geq0}(-1)^n
913 \frac{z^{Nn+d_{pq}}}{(Nn+d_{pq})!}.
914 \]
915 This is the stated formula.
916 \end{proof}
917 
918 \begin{corollary}[Addition and inversion laws for the companion matrix]\label{cor:companion_addition_inversion}
919 Assume $N\geq2$.  The matrix
920 \[
921 U_N(z)=e^{zJ_N}
922 \]
923 satisfies
924 \[
925 U_N(z+w)=U_N(z)U_N(w),
926 \qquad
927 U_N(z)^{-1}=U_N(-z),
928 \qquad
929 \det U_N(z)=1.
930 \]
931 Its first column is $\mathbf D_N(z)$, and all of its entries are given by \cref{cor:fundamental_matrix_entries}.
932 \end{corollary}
933 
934 \begin{proof}
935 The first two identities are the exponential group law.  Since $\tr J_N=0$,
936 \[
937 \det(e^{zJ_N})=e^{z\tr J_N}=1.
938 \]
939 The first column statement is \cref{thm:companion_matrix_realization}.
940 \end{proof}
941 
942 \clearpage\part{Reciprocal coefficients and polynomial families}
943 \section{Reciprocal powers and parameter polynomials}\label{sec:reciprocal_powers}
944 
945 \begin{readerroadmap}
946 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.
947 \end{readerroadmap}
948 \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}
949 
950 
951 Since $\Dfun_N(0)=1$, its formal logarithm is defined at the origin.  On a
952 simply connected zero-free region containing the origin, there is a unique
953 holomorphic branch normalized by $\log\Dfun_N(0)=0$, and its germ at the
954 origin agrees with the formal logarithm.
955 
956 \begin{definition}\label{def:reciprocal_family}
957 For $r\in\CC$, define
958 \[
959 \Sfun_N^{(r)}(z)
960 =
961 \Dfun_N(z)^{-r}
962 =
963 \exp\bigl(-r\log\Dfun_N(z)\bigr).
964 \]
965 Write
966 \[
967 \Sfun_N^{(r)}(z)
968 =
969 \sum_{n\geq0}P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}.
970 \]
971 Define the logarithmic coefficients $\kappa_{N,m}$ by
972 \[
973 -\log\Delta_N(w)
974 =
975 \sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}.
976 \]
977 \end{definition}
978 
979 The polynomial dependence on $r$ can be read directly from the formal logarithm.  We use the complete exponential Bell polynomials $B_n$ defined by
980 \[
981 \exp\left(\sum_{m\geq1}x_m\frac{t^m}{m!}\right)
982 =
983 \sum_{n\geq0}B_n(x_1,\ldots,x_n)\frac{t^n}{n!}.
984 \]
985 
986 \begin{theorem}[Bell polynomial formula]\label{thm:reciprocal_bell}
987 For every $n\geq0$,
988 \[
989 P_{N,n}(r)
990 =
991 \frac{(Nn)!}{n!}
992 B_n\bigl(
993  r\kappa_{N,1},
994  r\,1!\,\kappa_{N,2},
995  \ldots,
996  r\,(n-1)!\,\kappa_{N,n}
997 \bigr).
998 \]
999 For $n\geq1$, this is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is
1000 \[
1001 \frac{(Nn)!}{n!(N!)^n}.
1002 \]
1003 \end{theorem}
1004 
1005 \begin{proof}
1006 The identity
1007 \[
1008 \Delta_N(w)^{-r}
1009 =
1010 \exp\left(
1011  r\sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}
1012 \right)
1013 \]
1014 fits the defining generating function for $B_n$ with
1015 \[
1016 x_m=r(m-1)!\kappa_{N,m}.
1017 \]
1018 The coefficient of $w^n$ is therefore the displayed Bell polynomial divided by $n!$.  Multiplication by $(Nn)!$ gives $P_{N,n}(r)$.  Since
1019 \[
1020 \Delta_N(w)=1-\frac{w}{N!}+O(w^2),
1021 \]
1022 we have $\kappa_{N,1}=1/N!$.  The term $x_1^n$ gives the stated leading coefficient, which is nonzero.
1023 \end{proof}
1024 
1025 The coefficients also satisfy a triangular recurrence that does not require the logarithmic coefficients to be known first.
1026 
1027 \begin{theorem}[Parameter recurrence]\label{thm:parameter_recurrence}
1028 We have $P_{N,0}(r)=1$.  For $n\geq1$,
1029 \[
1030 \begin{aligned}
1031 P_{N,n}(r)
1032 ={}&-
1033 \sum_{k=1}^{n-1}
1034 (-1)^k
1035 \binom{Nn-1}{Nk}
1036 P_{N,n-k}(r)
1037 \\
1038 &-r
1039 \sum_{k=1}^{n}
1040 (-1)^k
1041 \binom{Nn-1}{Nk-1}
1042 P_{N,n-k}(r).
1043 \end{aligned}
1044 \]
1045 \end{theorem}
1046 
1047 \begin{proof}
1048 Set $S=\Dfun_N^{-r}$.  Logarithmic differentiation gives
1049 \[
1050 \Dfun_NS'+r\Dfun_N'S=0.
1051 \]
1052 Taking the coefficient of $z^{Nn-1}/(Nn-1)!$ gives
1053 \[
1054 \sum_{k=0}^{n-1}(-1)^k
1055 \binom{Nn-1}{Nk}P_{N,n-k}(r)
1056 +r\sum_{k=1}^{n}(-1)^k
1057 \binom{Nn-1}{Nk-1}P_{N,n-k}(r)=0.
1058 \]
1059 The term with $k=0$ in the first sum is $P_{N,n}(r)$, and every
1060 other term has a smaller block index.  Rearrangement gives the formula.
1061 \end{proof}
1062 
1063 \begin{theorem}[Additive parameter law for reciprocal powers]\label{thm:reciprocal_parameter_addition}
1064 For all $r,s\in\CC$,
1065 \[
1066 \Sfun_N^{(r)}(z)\Sfun_N^{(s)}(z)
1067 =
1068 \Sfun_N^{(r+s)}(z)
1069 \]
1070 as a formal identity and on every common zero-free region with a fixed logarithm.  Consequently,
1071 \[
1072 P_{N,n}(r+s)
1073 =
1074 \sum_{q=0}^{n}
1075 \binom{Nn}{Nq}
1076 P_{N,q}(r)P_{N,n-q}(s).
1077 \]
1078 \end{theorem}
1079 
1080 \begin{proof}
1081 Both statements follow by adding the exponents of the same logarithm and then multiplying the block exponential generating series.
1082 \end{proof}
1083 
1084 \begin{corollary}[A sequence of binomial type]\label{cor:binomial_type}
1085 Define
1086 \[
1087 A_{N,n}(r)
1088 =
1089 \frac{n!}{(Nn)!}P_{N,n}(r).
1090 \]
1091 Then
1092 \[
1093 A_{N,n}(r+s)
1094 =
1095 \sum_{q=0}^{n}
1096 \binom nq
1097 A_{N,q}(r)A_{N,n-q}(s).
1098 \]
1099 Thus $(A_{N,n})_{n\geq0}$ is a polynomial sequence of binomial type.
1100 \end{corollary}
1101 
1102 \begin{proof}
1103 Multiply the coefficient identity in \cref{thm:reciprocal_parameter_addition} by $n!/(Nn)!$ and simplify the factorials.
1104 \end{proof}
1105 
1106 
1107 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.
1108 
1109 \begin{theorem}[Logarithmic Newton recurrence]\label{thm:logarithmic_newton_recurrence}
1110 Set
1111 \[
1112 b_{N,n}(r)=\frac{P_{N,n}(r)}{(Nn)!}.
1113 \]
1114 Then $b_{N,0}(r)=1$ and
1115 for every $n\geq1$,
1116 \[
1117 nb_{N,n}(r)
1118 =
1119 r\sum_{k=1}^{n}\kappa_{N,k}b_{N,n-k}(r).
1120 \]
1121 Equivalently,
1122 \[
1123 P_{N,n}(r)
1124 =
1125 \frac r n
1126 \sum_{k=1}^{n}
1127 \frac{(Nn)!}{(N(n-k))!}
1128 \kappa_{N,k}P_{N,n-k}(r).
1129 \]
1130 If $p_k=r\kappa_{N,k}$, then, for $n\geq1$,
1131 \[
1132 b_{N,n}(r)
1133 =
1134 \frac1{n!}
1135 \det
1136 \begin{pmatrix}
1137 p_1&-1&0&\cdots&0\\
1138 p_2&p_1&-2&\cdots&0\\
1139 p_3&p_2&p_1&\ddots&\vdots\\
1140 \vdots&\vdots&\ddots&\ddots&-(n-1)\\
1141 p_n&p_{n-1}&\cdots&p_2&p_1
1142 \end{pmatrix}.
1143 \]
1144 \end{theorem}
1145 
1146 \begin{proof}
1147 Let
1148 \[
1149 B_r(w)=\Delta_N(w)^{-r}
1150 =
1151 \sum_{n\geq0}b_{N,n}(r)w^n.
1152 \]
1153 Logarithmic differentiation gives
1154 \[
1155 \frac{B_r'(w)}{B_r(w)}
1156 =
1157 r\sum_{k\geq1}\kappa_{N,k}w^{k-1}.
1158 \]
1159 Comparison of the coefficient of $w^{n-1}$ proves the recurrence.
1160 
1161 Let $D_n$ denote the displayed determinant, with $D_0=1$.  Expansion along the last row gives
1162 \[
1163 D_n
1164 =
1165 \sum_{k=1}^{n}
1166 \frac{(n-1)!}{(n-k)!}p_kD_{n-k}.
1167 \]
1168 Here the cofactor sign is canceled by the product of the negative
1169 superdiagonal entries $-(n-k+1),\ldots,-(n-1)$.
1170 Thus $D_n/n!$ satisfies the same recurrence and initial value as $b_{N,n}(r)$.
1171 \end{proof}
1172 
1173 \begin{corollary}[The first three reciprocal polynomials]\label{cor:first_reciprocal_polynomials}
1174 For every $N\geq1$,
1175 \[
1176 P_{N,1}(r)=r,
1177 \]
1178 \[
1179 P_{N,2}(r)
1180 =
1181 r\left[
1182 \frac12\binom{2N}{N}(r+1)-1
1183 \right],
1184 \]
1185 and
1186 \[
1187 P_{N,3}(r)
1188 =
1189 r\left[
1190 \frac{(3N)!}{6(N!)^3}(r+1)(r+2)
1191 -
1192 \binom{3N}{N}(r+1)
1193 +1
1194 \right].
1195 \]
1196 \end{corollary}
1197 
1198 \begin{proof}
1199 The first logarithmic coefficients are
1200 \[
1201 \kappa_{N,1}=\frac1{N!},
1202 \]
1203 \[
1204 \kappa_{N,2}=\frac1{(N!)^2}-\frac2{(2N)!},
1205 \]
1206 and
1207 \[
1208 \kappa_{N,3}
1209 =
1210 \frac1{(N!)^3}
1211 -
1212 \frac3{N!(2N)!}
1213 +
1214 \frac3{(3N)!}.
1215 \]
1216 Insert these values into \cref{thm:reciprocal_bell} and simplify.
1217 \end{proof}
1218 
1219 \begin{corollary}\label{cor:factor_r}
1220 For $n\geq1$, the polynomial $P_{N,n}(r)$ is divisible by $r$.
1221 \end{corollary}
1222 
1223 \begin{proof}
1224 At $r=0$, the reciprocal family is the constant function one.  Every positive-index block coefficient therefore vanishes.
1225 \end{proof}
1226 
1227 \begin{example}\label{ex:reciprocal_polynomials}
1228 For $N=1$, one has $\Dfun_1(z)^{-r}=e^{rz}$ and $P_{1,n}(r)=r^n$.  For $N=2$,
1229 \[
1230 P_{2,1}(r)=r,
1231 \]
1232 \[
1233 P_{2,2}(r)=r(3r+2),
1234 \]
1235 and
1236 \[
1237 P_{2,3}(r)=r(15r^2+30r+16).
1238 \]
1239 At $r=1$, these are the positive secant coefficients.
1240 \end{example}
1241 
1242 \section{Logarithmic derivatives and Riccati equations}\label{sec:tangent}
1243 
1244 \begin{microexample}[title={A coefficient calculation at order two}]
1245 Write $\Delta_2(w)^{-r}=\sum_{n\geq0}b_n(r)w^n$. Since
1246 \[
1247 \Delta_2(w)=1-\frac w2+\frac{w^2}{24}+O(w^3),
1248 \qquad -\log\Delta_2(w)=\frac w2+\frac{w^2}{12}+O(w^3),
1249 \]
1250 the first logarithmic coefficients are $\kappa_{2,1}=1/2$ and $\kappa_{2,2}=1/6$. Newton's recurrence gives
1251 \[
1252 b_1(r)=\frac r2,\qquad
1253 2b_2(r)=r\left(\frac r4+\frac16\right).
1254 \]
1255 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.
1256 \end{microexample}
1257 \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}
1258 
1259 
1260 \begin{definition}\label{def:transfer_tangent}
1261 Define the transfer tangent
1262 \[
1263 \Tfun_N(z)
1264 =-\frac{\Dfun_N'(z)}{\Dfun_N(z)}.
1265 \]
1266 Near the origin, its convergent block expansion has the form
1267 \[
1268 \Tfun_N(z)
1269 =
1270 \sum_{n\geq0}T_N(n)
1271 \frac{z^{Nn+N-1}}{(Nn+N-1)!}.
1272 \]
1273 \end{definition}
1274 
1275 \begin{proposition}[Logarithmic coefficient formula]\label{prop:tangent_kappa}
1276 The transfer tangent satisfies
1277 \[
1278 \Tfun_N(z)
1279 =
1280 N\sum_{m\geq1}\kappa_{N,m}z^{Nm-1}.
1281 \]
1282 Equivalently,
1283 \[
1284 T_N(n)
1285 =
1286 N(Nn+N-1)!\kappa_{N,n+1}.
1287 \]
1288 \end{proposition}
1289 
1290 \begin{proof}
1291 Since $\Dfun_N(z)=\Delta_N(z^N)$,
1292 \[
1293 -\frac{\Dfun_N'(z)}{\Dfun_N(z)}
1294 =
1295 -Nz^{N-1}\frac{\Delta_N'(z^N)}{\Delta_N(z^N)}.
1296 \]
1297 Differentiate the defining series for $-\log\Delta_N$ and compare coefficients.
1298 \end{proof}
1299 
1300 \begin{theorem}[Tangent recurrence]\label{thm:tangent_recurrence}
1301 The coefficients satisfy $T_N(0)=1$ and
1302 \[
1303 T_N(n)
1304 =
1305 (-1)^n
1306 -
1307 \sum_{k=1}^{n}
1308 (-1)^k
1309 \binom{Nn+N-1}{Nk}
1310 T_N(n-k).
1311 \]
1312 \end{theorem}
1313 
1314 \begin{proof}
1315 The defining equation is
1316 \[
1317 \Dfun_N\Tfun_N=-\Dfun_N'.
1318 \]
1319 Extracting the coefficient of $z^{Nn+N-1}/(Nn+N-1)!$ gives
1320 \[
1321 \sum_{k=0}^{n}(-1)^k
1322 \binom{Nn+N-1}{Nk}T_N(n-k)=(-1)^n.
1323 \]
1324 Isolating the $k=0$ term gives the recurrence.
1325 \end{proof}
1326 
1327 \begin{corollary}[Derivative of the reciprocal family]\label{cor:reciprocal_derivative}
1328 For every $r\in\CC$, the formal identity
1329 \[
1330 \frac{d}{dz}\Sfun_N^{(r)}(z)
1331 =
1332 r\Tfun_N(z)\Sfun_N^{(r)}(z)
1333 \]
1334 holds.  The same identity holds analytically on every zero-free
1335 domain on which the logarithm used to define $\Sfun_N^{(r)}$ is fixed.
1336 \end{corollary}
1337 
1338 \begin{proof}
1339 Differentiate $\Dfun_N^{-r}$ on a zero-free region.  The formal identity follows from the same logarithmic calculation.
1340 \end{proof}
1341 
1342 For a function $g$, define the complete differential Bell polynomial $Y_N$ by
1343 \[
1344 \frac{d^N}{dz^N}e^{g(z)}
1345 =
1346 e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr).
1347 \]
1348 
1349 \begin{theorem}[Scalar higher Riccati identity]\label{thm:higher_riccati}
1350 At every point of the open set $\{z:\Dfun_N(z)\ne0\}$,
1351 \[
1352 Y_N\bigl(
1353 -\Tfun_N,
1354 -\Tfun_N',
1355 \ldots,
1356 -\Tfun_N^{(N-1)}
1357 \bigr)
1358 =-1.
1359 \]
1360 \end{theorem}
1361 
1362 \begin{proof}
1363 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 $Y_N$.
1364 \end{proof}
1365 
1366 This scalar identity follows from one ordinary differential equation. It does
1367 not redevelop the general nonlinear operation-tree theory of Chapter III.
1368 
1369 \begin{example}\label{ex:ordinary_tangent}
1370 For $N=1$, $\Tfun_1=1$.  For $N=2$, $\Dfun_2=\cos z$ and $\Tfun_2=\tan z$.  The Riccati equation becomes
1371 \[
1372 \Tfun_2'=1+\Tfun_2^2.
1373 \]
1374 \end{example}
1375 
1376 \section{Residue-section reciprocals and Appell systems}\label{sec:residue_section}
1377 
1378 The section $\Efun_{N,j}$ has a forced zero of order $j$ at the origin.  Removing it leaves a unit series in $z^N$.
1379 
1380 \begin{definition}\label{def:normalized_section}
1381 For $0\leq j<N$, define
1382 \[
1383 \Phi_{N,j}(w)
1384 =
1385 j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!}.
1386 \]
1387 Then
1388 \[
1389 \Efun_{N,j}(z)
1390 =
1391 \frac{z^j}{j!}\Phi_{N,j}(z^N).
1392 \]
1393 Define the residue-section reciprocal function
1394 \[
1395 \Cfun_{N,j}(z)
1396 =
1397 \Phi_{N,j}(z^N)^{-1}
1398 =
1399 \frac{z^j/j!}{\Efun_{N,j}(z)}.
1400 \]
1401 For $j>0$, the last quotient at $z=0$ is interpreted by its removable
1402 continuation, namely the first expression in the display.
1403 Write
1404 \[
1405 \Cfun_{N,j}(z)
1406 =
1407 \sum_{n\geq0}C_{N,j}(n)\frac{z^{Nn}}{(Nn)!}.
1408 \]
1409 \end{definition}
1410 
1411 \begin{theorem}[Triangular inverse system]\label{thm:triangular_inverse}
1412 The coefficients are determined by $C_{N,j}(0)=1$ and
1413 \[
1414 \sum_{q=0}^{n}
1415 \binom{Nn+j}{Nq}
1416 C_{N,j}(q)
1417 =
1418 \delta_{n0}.
1419 \]
1420 For $n\geq1$,
1421 \[
1422 C_{N,j}(n)
1423 =
1424 -
1425 \frac{
1426 \displaystyle\sum_{q=0}^{n-1}
1427 \binom{Nn+j}{Nq}C_{N,j}(q)
1428 }{
1429 \displaystyle\binom{Nn+j}{Nn}
1430 }.
1431 \]
1432 \end{theorem}
1433 
1434 \begin{proof}
1435 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.
1436 \end{proof}
1437 
1438 \begin{definition}\label{def:residue_section_power}
1439 For $r\in\CC$, define formal coefficients $C_{N,j,n}(r)$ by
1440 \[
1441 \Cfun_{N,j}(z)^r
1442 =
1443 \sum_{n\geq0}C_{N,j,n}(r)\frac{z^{Nn}}{(Nn)!}.
1444 \]
1445 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 $L(0)=0$.
1446 \end{definition}
1447 
1448 \begin{theorem}[Parameter-addition law for residue-section reciprocals]\label{thm:residue_section_parameter_addition}
1449 For each $n\geq1$, the coefficient $C_{N,j,n}(r)$ is a polynomial in $r$ of exact degree $n$.  Its leading coefficient is
1450 \[
1451 \frac{(Nn)!}{n!}
1452 \left(-\frac{j!}{(N+j)!}\right)^n.
1453 \]
1454 It satisfies
1455 \[
1456 C_{N,j,n}(r+s)
1457 =
1458 \sum_{q=0}^{n}
1459 \binom{Nn}{Nq}
1460 C_{N,j,q}(r)C_{N,j,n-q}(s).
1461 \]
1462 In particular,
1463 \[
1464 C_{N,j,n}(1)=C_{N,j}(n).
1465 \]
1466 \end{theorem}
1467 
1468 \begin{proof}
1469 The unit series $\Phi_{N,j}(w)^{-1}$ has a formal logarithm with zero constant term.  Its linear term is
1470 \[
1471 -\frac{j!}{(N+j)!}w.
1472 \]
1473 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.
1474 More explicitly, before the factorial normalization the coefficient of
1475 $r^nw^n$ is
1476 \[
1477 \frac1{n!}\left(-\frac{j!}{(N+j)!}\right)^n;
1478 \]
1479 thus multiplication by $(Nn)!$ gives the displayed leading coefficient.
1480 \end{proof}
1481 
1482 \begin{definition}\label{def:block_appell}
1483 Define polynomials $E_m^{(N,j,r)}(x)$ by
1484 \[
1485 \Cfun_{N,j}(z)^re^{xz}
1486 =
1487 \sum_{m\geq0}E_m^{(N,j,r)}(x)\frac{z^m}{m!}.
1488 \]
1489 \end{definition}
1490 
1491 \begin{theorem}[Appell laws]\label{thm:appell_laws}
1492 For $m\geq1$, the polynomials satisfy
1493 \[
1494 \frac{d}{dx}E_m^{(N,j,r)}(x)
1495 =
1496 mE_{m-1}^{(N,j,r)}(x)
1497 \]
1498 and, for $m\geq0$,
1499 \[
1500 E_m^{(N,j,r)}(x+y)
1501 =
1502 \sum_{q=0}^{m}
1503 \binom mq
1504 E_q^{(N,j,r)}(x)y^{m-q}.
1505 \]
1506 \end{theorem}
1507 
1508 \begin{proof}
1509 Differentiate the generating function with respect to $x$ for the first identity.  Multiply it by $e^{yz}$ and compare coefficients for the second.
1510 \end{proof}
1511 
1512 \begin{proposition}[Explicit Appell polynomial]\label{prop:explicit_appell}
1513 For every $m\geq0$,
1514 \[
1515 E_m^{(N,j,r)}(x)
1516 =
1517 \sum_{0\leq n\leq m/N}
1518 \binom{m}{Nn}
1519 C_{N,j,n}(r)x^{m-Nn}.
1520 \]
1521 \end{proposition}
1522 
1523 \begin{proof}
1524 Multiply the block series for $\Cfun_{N,j}^r$ by the exponential series and collect the coefficient of total degree $m$.
1525 \end{proof}
1526 
1527 
1528 \begin{theorem}[Joint Appell convolution in the space and order parameters]\label{thm:appell_two_parameter_convolution}
1529 For all $r,s,x,y\in\CC$ and $m\geq0$,
1530 \[
1531 E_m^{(N,j,r+s)}(x+y)
1532 =
1533 \sum_{q=0}^{m}
1534 \binom mq
1535 E_q^{(N,j,r)}(x)
1536 E_{m-q}^{(N,j,s)}(y).
1537 \]
1538 \end{theorem}
1539 
1540 \begin{proof}
1541 Multiply the generating functions
1542 \[
1543 \Cfun_{N,j}(z)^re^{xz}
1544 \quad\hbox{and}\quad
1545 \Cfun_{N,j}(z)^se^{yz}.
1546 \]
1547 Their product is the generating function with parameters $r+s$ and $x+y$.  Coefficient comparison gives the formula.
1548 \end{proof}
1549 
1550 \begin{proposition}[Formal differential-operator representation on polynomials]\label{prop:appell_operator_form}
1551 Let a formal series $A(z)=\sum_{q\geq0}a_qz^q/q!$ act on polynomials by
1552 \[
1553 A(\partial_x)f(x)
1554 =
1555 \sum_{q\geq0}\frac{a_q}{q!}\partial_x^qf(x).
1556 \]
1557 The power $r$ is applied to the formal symbol before the substitution $z=\partial_x$.  Then
1558 \[
1559 E_m^{(N,j,r)}(x)
1560 =
1561 \Cfun_{N,j}(\partial_x)^r x^m.
1562 \]
1563 The inverse relation is
1564 \[
1565 x^m
1566 =
1567 \Phi_{N,j}(\partial_x^N)^r
1568 E_m^{(N,j,r)}(x).
1569 \]
1570 Both sums are finite on every polynomial.
1571 \end{proposition}
1572 
1573 \begin{proof}
1574 Applying $A(\partial_x)$ to $e^{xz}$ multiplies it by $A(z)$.  Taking
1575 $A=\Cfun_{N,j}^r$ gives the first identity by the defining generating
1576 function.  Constant coefficient operators on polynomials satisfy
1577 \[
1578 (AB)(\partial_x)=A(\partial_x)B(\partial_x),
1579 \]
1580 because all derivative sums are finite and powers of $\partial_x$
1581 commute.  The two formal symbols
1582 \[
1583 \Cfun_{N,j}(z)^r=\Phi_{N,j}(z^N)^{-r}
1584 \quad\hbox{and}\quad
1585 \Phi_{N,j}(z^N)^r
1586 \]
1587 multiply to one.  Their substitutions at $z=\partial_x$ are therefore
1588 inverse constant coefficient operators on polynomials, which proves
1589 the second identity.
1590 \end{proof}
1591 
1592 
1593 \begin{corollary}[Appell symmetry]\label{cor:appell_symmetry}
1594 Each polynomial $E_m^{(N,j,r)}$ is monic of degree $m$, and
1595 \[
1596 E_m^{(N,j,r)}(\omega x)
1597 =
1598 \omega^mE_m^{(N,j,r)}(x).
1599 \]
1600 \end{corollary}
1601 
1602 \begin{proof}
1603 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 $\omega^m$.
1604 \end{proof}
1605 
1606 \begin{example}\label{ex:euler_boundaries}
1607 For $(N,j)=(2,0)$,
1608 \[
1609 \Cfun_{2,0}(z)=\sech z.
1610 \]
1611 For $(N,j)=(2,1)$,
1612 \[
1613 \Cfun_{2,1}(z)=\frac z{\sinh z}.
1614 \]
1615 The first case is a scaled and translated Euler family.  In fact, if $E_m(x)$ denotes the classical Euler polynomial, then
1616 \[
1617 E_m^{(2,0,1)}(x)
1618 =
1619 2^mE_m\left(\frac{x+1}{2}\right).
1620 \]
1621 If $B_m(x)$ denotes the classical Bernoulli polynomial, the second case has
1622 the exact identification
1623 \[
1624 E_m^{(2,1,1)}(x)
1625 =
1626 2^mB_m\left(\frac{x+1}{2}\right),
1627 \]
1628 obtained by substituting $t=2z$ in
1629 $t e^{yt}/(e^t-1)$.
1630 \end{example}
1631 
1632 \section{Scaled reciprocal coupling and theta polynomials}\label{sec:theta}
1633 
1634 \begin{definition}\label{def:unsigned_denominator}
1635 Set
1636 \[
1637 H_N(z)=\Efun_{N,0}(z)
1638 =
1639 \sum_{n\geq0}\frac{z^{Nn}}{(Nn)!}.
1640 \]
1641 For $u\in\CC$, define $\Thetafun_{N,n}(u)$ by
1642 \[
1643 H_N(uz)^{-1}H_N(z)^{-u}
1644 =
1645 \sum_{n\geq0}\Thetafun_{N,n}(u)\frac{z^{Nn}}{(Nn)!}.
1646 \]
1647 This is first a formal identity at the origin.  Analytically, it is interpreted
1648 on a common simply connected zero-free region with the branch of
1649 $\log H_N$ normalized by $\log H_N(0)=0$.
1650 \end{definition}
1651 
1652 \begin{theorem}[Theta reconstruction]\label{thm:theta_reconstruction}
1653 For every $n\geq0$,
1654 \[
1655 \Thetafun_{N,n}(u)
1656 =
1657 (-1)^n
1658 \sum_{q=0}^{n}
1659 \binom{Nn}{Nq}
1660 P_{N,q}(1)u^{Nq}P_{N,n-q}(u).
1661 \]
1662 Thus $\Thetafun_{N,n}$ is a polynomial of degree at most $Nn$.
1663 \end{theorem}
1664 
1665 \begin{proof}
1666 Choose $\rho$ with $\rho^N=-1$.  Then
1667 \[
1668 H_N(z)=\Dfun_N(\rho z).
1669 \]
1670 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.
1671 \end{proof}
1672 
1673 \begin{proposition}[Trivial parameter factors]\label{prop:theta_factors}
1674 For $n\geq1$, the polynomial $\Thetafun_{N,n}(u)$ is divisible by $u$.  If $N$ is even, it is also divisible by $u+1$.
1675 \end{proposition}
1676 
1677 \begin{proof}
1678 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.
1679 \end{proof}
1680 
1681 
1682 \begin{proposition}[Theta Newton recurrence]\label{prop:theta_newton_recurrence}
1683 Put
1684 \[
1685 \theta_{N,n}(u)=\frac{\Thetafun_{N,n}(u)}{(Nn)!}.
1686 \]
1687 Then $\theta_{N,0}(u)=1$ and
1688 \[
1689 n\theta_{N,n}(u)
1690 =
1691 \sum_{m=1}^{n}
1692 (-1)^m\kappa_{N,m}
1693 \bigl(u^{Nm}+u\bigr)
1694 \theta_{N,n-m}(u).
1695 \]
1696 In particular,
1697 \[
1698 \Thetafun_{N,1}(u)=-(u^N+u).
1699 \]
1700 \end{proposition}
1701 
1702 \begin{proof}
1703 Write $w=z^N$.  Since $H_N(z)=\Delta_N(-w)$,
1704 \[
1705 -\log H_N(z)
1706 =
1707 \sum_{m\geq1}(-1)^m\kappa_{N,m}\frac{w^m}{m}.
1708 \]
1709 The logarithm of the theta generating function is therefore
1710 \[
1711 \sum_{m\geq1}
1712 (-1)^m\kappa_{N,m}
1713 \bigl(u^{Nm}+u\bigr)
1714 \frac{w^m}{m}.
1715 \]
1716 Logarithmic differentiation and coefficient comparison give the recurrence.  The case $n=1$ uses $\kappa_{N,1}=1/N!$.
1717 \end{proof}
1718 
1719 \begin{example}\label{ex:theta_even}
1720 For $N=2$,
1721 \[
1722 \Thetafun_{2,1}(u)=-u(u+1)
1723 \]
1724 and
1725 \[
1726 \Thetafun_{2,2}(u)
1727 =
1728 u(u+1)(5u^2+u+2).
1729 \]
1730 The two factors predicted by \cref{prop:theta_factors} are visible.
1731 \end{example}
1732 
1733 \clearpage\part{The geometry of zeros}
1734 \section{Growth and the geometry of the large zeros}\label{sec:zero_geometry}
1735 
1736 \begin{readerroadmap}
1737 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.
1738 \end{readerroadmap}
1739 \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}
1740 
1741 
1742 We now assume $N\geq2$.  Write
1743 \[
1744 \eta_k=e^{(2k+1)\pi i/N},
1745 \qquad 0\leq k<N.
1746 \]
1747 These are the vertices of a regular polygon and
1748 \[
1749 \Dfun_N(z)=\frac1N\sum_{k=0}^{N-1}e^{\eta_kz}.
1750 \]
1751 
1752 \begin{standardfact}[Order from Taylor coefficients]\label{sf:coefficient_order}
1753 If $f(w)=\sum_{n\geq0}a_nw^n$ is an entire function with infinitely many nonzero coefficients, then
1754 \[
1755 \ord f
1756 =
1757 \limsup_{\substack{n\to\infty\\a_n\ne0}}
1758 \frac{n\log n}{\log(1/|a_n|)}.
1759 \]
1760 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}.
1761 \end{standardfact}
1762 
1763 \begin{proposition}[Orders and exponential type]\label{prop:orders}
1764 For $N\geq2$,
1765 \[
1766 \ord\Delta_N=\frac1N.
1767 \]
1768 The function $\Dfun_N$ has order one and exponential type one.
1769 \end{proposition}
1770 
1771 \begin{proof}
1772 The coefficient of $w^n$ in $\Delta_N$ has modulus $1/(Nn)!$.
1773 Stirling's formula and \cref{sf:coefficient_order} give order $1/N$.
1774 Since $M_{f(\,\cdot\,^N)}(r)=M_f(r^N)$ for the maximum modulus,
1775 composition with $z^N$ multiplies the order by $N$ and gives order one
1776 for $\Dfun_N$.
1777 
1778 The finite exponential representation gives
1779 \[
1780 |\Dfun_N(z)|\leq e^{|z|},
1781 \]
1782 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
1783 \[
1784 \Dfun_N(t\overline{\eta_0})
1785 =
1786 \frac1Ne^t(1+o(1)).
1787 \]
1788 The type is therefore one.
1789 \end{proof}
1790 
1791 
1792 \begin{definition}\label{def:indicator_polygon}
1793 For $\theta\in\mathbb R$, define
1794 \[
1795 h_N(\theta)
1796 =
1797 \max_{0\leq k<N}
1798 \cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
1799 \]
1800 The tie directions are
1801 \[
1802 \mathfrak T_N
1803 =
1804 \left\{\frac{2\pi k}{N}:0\leq k<N\right\}
1805 \quad\bmod 2\pi.
1806 \]
1807 \end{definition}
1808 
1809 \begin{theorem}[Indicator polygon and dominant sectors]\label{thm:indicator_polygon}
1810 The indicator of $\Dfun_N$ equals $h_N$, with the limsup taken over radii where $\Dfun_N(re^{i\theta})\ne0$:
1811 \[
1812 \limsup_{r\to\infty}
1813 \frac1r\log|\Dfun_N(re^{i\theta})|
1814 =
1815 h_N(\theta).
1816 \]
1817 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
1818 \[
1819 \Dfun_N(re^{i\theta})
1820 =
1821 \frac1N
1822 \exp\bigl(\eta(\theta)re^{i\theta}\bigr)
1823 \left(1+O(e^{-\delta r})\right)
1824 \]
1825 uniformly for $r\geq R$.  In particular, no sufficiently large zero can lie in those directions.
1826 \end{theorem}
1827 
1828 \begin{proof}
1829 For $z=re^{i\theta}$, the real part of the exponent $\eta_kz$ is
1830 \[
1831 r\cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
1832 \]
1833 The triangle inequality gives the upper indicator bound $h_N(\theta)$.
1834 
1835 Away from a tie direction, the maximizing vertex is unique.  It is constant
1836 on each connected component of the complement of the tie directions.  A
1837 compact set avoiding all ties meets only finitely many such components, and
1838 compactness gives one uniform positive lower bound for every relevant
1839 projection gap.  Division by the maximizing exponential gives the stated
1840 uniform asymptotic and the zero-free conclusion.
1841 
1842 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
1843 \[
1844 \frac2N e^{c_Nr}\cos(s_Nr)
1845 \]
1846 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.
1847 \end{proof}
1848 
1849 The zero strings are governed by two adjacent exponents.  Put
1850 \[
1851 a_N=\frac\pi N,
1852 \qquad
1853 s_N=\sin a_N,
1854 \qquad
1855 c_N=\cos a_N,
1856 \]
1857 and
1858 \[
1859 A_N=\frac\pi{s_N},
1860 \qquad
1861  t_m=A_N\left(m+\frac12\right).
1862 \]
1863 For $N\geq3$, also put
1864 \[
1865 \gamma_N=c_N-\cos(3a_N)>0.
1866 \]
1867 
1868 \begin{lemma}[Two term dominance]\label{lem:two_term_dominance}
1869 Assume $N\geq3$.  There are positive constants $Y$, $R$, and $C$ with the following properties.
1870 
1871 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 $|y|\leq Y$.
1872 
1873 On the strip $x\geq R$ and $|y|\leq Y$, define
1874 \[
1875 R_N(z)
1876 =
1877 \sum_{\substack{\eta^N=-1\\
1878 \eta\ne e^{ia_N},e^{-ia_N}}}
1879 \exp\bigl((\eta-e^{-ia_N})z\bigr).
1880 \]
1881 Then
1882 \[
1883 N e^{-e^{-ia_N}z}\Dfun_N(z)
1884 =
1885 1+e^{2is_Nz}+R_N(z),
1886 \]
1887 and, for $q=0,1,2$,
1888 \[
1889 |R_N^{(q)}(z)|
1890 \leq
1891 C e^{-\gamma_Nx}.
1892 \]
1893 \end{lemma}
1894 
1895 \begin{proof}
1896 A direct projection estimate gives the stronger bound
1897 \[
1898 Y=\frac{\log(N-1)}{2s_N}
1899 \]
1900 for every zero in the sector, without a large-radius assumption.
1901 Write $z=re^{i\theta}$ with $0\leq\theta\leq a_N$ and $r\geq0$.
1902 For $0\leq\ell\leq N-2$, the angle
1903 $\theta+(2\ell+1)a_N$ lies between $a_N+\theta$ and
1904 $2\pi-a_N-\theta$. Monotonicity of cosine on $[0,\pi]$, followed by
1905 reflection about $\pi$, therefore gives
1906 \[
1907 \Re(\eta_\ell z)\leq r\cos(a_N+\theta).
1908 \]
1909 The remaining vertex $e^{-ia_N}$ has projection $r\cos(a_N-\theta)$.
1910 Their difference is at least $2s_Nr\sin\theta=2s_Ny$.
1911 If $\Dfun_N(z)=0$, divide its finite exponential sum by the term
1912 with exponent $e^{-ia_N}z$ and apply the triangle inequality:
1913 \[
1914 1\leq(N-1)e^{-2s_Ny}.
1915 \]
1916 Thus $y\leq Y$. Conjugation gives the same bound for $-y$ on the lower
1917 half sector. Since $N\geq3$ and $s_N>0$, this $Y$ is positive.
1918 
1919 On the resulting strip, the roots $e^{ia_N}$ and $e^{-ia_N}$ have
1920 real part $c_N$, while each remaining root $\eta=u+iv$ satisfies
1921 $u\leq\cos(3a_N)=c_N-\gamma_N$. Hence
1922 \[
1923 \Re\bigl((\eta-e^{-ia_N})z\bigr)
1924 \leq-\gamma_Nx+2Y.
1925 \]
1926 Division by $e^{e^{-ia_N}z}$ gives the displayed normalization.
1927 Each differentiation multiplies a remainder term by
1928 $\eta-e^{-ia_N}$, whose modulus is at most two. Thus, for $q=0,1,2$,
1929 \[
1930 |R_N^{(q)}(z)|\leq(N-2)2^qe^{2Y}e^{-\gamma_Nx}.
1931 \]
1932 One may take $R=1$ and $C=4(N-2)e^{2Y}$.
1933 \end{proof}
1934 
1935 \begin{lemma}[Near cancellation of two exponentials]\label{lem:near_cancellation}
1936 For every $M>0$, there are constants $\varepsilon_M>0$ and $C_M>0$ such that the following holds.  If
1937 \[
1938 |\Re w|\leq M
1939 \]
1940 and
1941 \[
1942 |1+e^w|\leq\varepsilon_M,
1943 \]
1944 then there is a unique integer $m$ with
1945 \[
1946 |w-(2m+1)\pi i|
1947 \leq
1948 C_M|1+e^w|.
1949 \]
1950 \end{lemma}
1951 
1952 \begin{proof}
1953 In fact, the constants $\varepsilon_M=1/2$ and $C_M=3/2$ work
1954 for every complex $w$, without the strip restriction. Put
1955 $u=-(1+e^w)$ and $L=\log(1+u)=\log(-e^w)$, using the principal
1956 logarithm. The elementary logarithm estimate on $|u|\leq1/2$ gives
1957 $|L|\leq(3/2)|u|$. Since $e^{L+\pi i}=e^w$, there is an integer
1958 $m$ such that
1959 \[
1960 w=L+(2m+1)\pi i.
1961 \]
1962 This gives the required distance bound. Distinct centers are separated
1963 by at least $2\pi$, whereas two centers satisfying the bound would
1964 be separated by at most $3|1+e^w|\leq3/2$. Thus the integer is unique.
1965 The formal proof uses the convergent logarithm estimate and the exact
1966 periodicity theorem for the complex exponential.
1967 \end{proof}
1968 
1969 \begin{theorem}[Large zero strings]\label{thm:large_zero_strings}
1970 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
1971 \[
1972 \tau_m
1973 =
1974  t_m+O(e^{-\gamma t_m}).
1975 \]
1976 Every zero of sufficiently large modulus is of the form
1977 \[
1978 \omega^k\tau_m
1979 \]
1980 for a unique $0\leq k<N$ and a unique $m\geq m_0$.
1981 
1982 The derivatives satisfy
1983 \[
1984 \Dfun_N'(\omega^k\tau_m)
1985 =
1986 \omega^{-k}\Dfun_N'(\tau_m)
1987 \]
1988 and
1989 \[
1990 |\Dfun_N'(\tau_m)|
1991 =
1992 \frac{2s_N}{N}e^{c_N\tau_m}
1993 \left(1+O(e^{-\gamma\tau_m})\right).
1994 \]
1995 \end{theorem}
1996 
1997 \begin{proof}
1998 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
1999 \[
2000 G(z)=N e^{-e^{-ia_N}z}\Dfun_N(z).
2001 \]
2002 Then
2003 \[
2004 G(z)=1+e^{2is_Nz}+R_N(z)
2005 \]
2006 with an exponentially small remainder and derivative.
2007 
2008 If $G(z)=0$, then
2009 \[
2010 |1+e^{2is_Nz}|
2011 \leq
2012 C e^{-\gamma_N\Re z}.
2013 \]
2014 The real part of $2is_Nz$ is bounded in the strip.  Apply
2015 \cref{lem:near_cancellation} with $w=2is_Nz$.  There is a unique
2016 integer $m$ such that
2017 \[
2018 |z-t_m|\leq C_1e^{-\gamma_N\Re z},
2019 \]
2020 where
2021 \[
2022  t_m=\frac{(2m+1)\pi}{2s_N}.
2023 \]
2024 In particular, $|\Re z-t_m|=o(1)$.  For all sufficiently large zeros
2025 we have $|\Re z-t_m|\leq1$, and hence
2026 \[
2027 |z-t_m|\leq C_2e^{-\gamma_Nt_m}
2028 \]
2029 with a constant independent of $m$.
2030 Because $\Re z\to+\infty$ in this argument, the corresponding integers
2031 $m$ are eventually nonnegative and tend to infinity.
2032 
2033 Fix a large $m$, put $a=2is_N$, and consider the closed disk
2034 $B_m=\{z:|z-t_m|\leq 1/(2s_N)\}$. The remainder bounds hold
2035 uniformly on these fixed-radius disks. For sufficiently large $m$,
2036 $|R_N|\leq 1/2$ and $|R_N'|\leq s_N/2$ there. Define
2037 \[
2038 \Phi_m(z)=t_m+\frac{\log(1+R_N(z))}{2is_N},
2039 \]
2040 using the principal logarithm. The estimate
2041 $|\log(1+u)|\leq(3/2)|u|$ shows that $\Phi_m$ maps $B_m$ into
2042 itself. The derivative bound $|\Phi_m'|\leq1/2$ makes it a contraction.
2043 Banach's fixed-point theorem gives a unique fixed point.
2044 On $B_m$, $|a(z-t_m)|\leq1<\pi$, so the principal logarithm inverts
2045 the exponential and the fixed-point equation is equivalent to $G(z)=0$.
2046 Moreover,
2047 \[
2048 |z-t_m|\leq\frac{3}{4s_N}|R_N(z)|=O(e^{-\gamma_Nt_m}).
2049 \]
2050 At this zero,
2051 \[
2052 |G'(z)|\geq |a|\,|1+R_N(z)|-|R_N'(z)|\geq s_N/2>0,
2053 \]
2054 so the zero is simple. The disks are mutually disjoint because their
2055 diameter $1/s_N$ is smaller than the center spacing $\pi/s_N$.
2056 Enlarging the strip if necessary, these disks lie inside the fundamental
2057 sector and the fixed strip for large $m$.
2058 
2059 The localization estimate puts every sufficiently large zero in the
2060 fundamental sector into one of these disks.  The zero sets of $G$ and
2061 $\Dfun_N$ coincide because their prefactor is nonzero.  Each disk is
2062 invariant under complex conjugation, and $\Dfun_N$ has real
2063 coefficients.  The conjugate of its unique zero is therefore the same
2064 zero.  It is real, positive for large $m$, and simple by the
2065 derivative bound above.  Denote it by $\tau_m$.
2066 
2067 The parts of the fundamental sector away from the positive tie ray,
2068 including its boundary rays, were already excluded uniformly.
2069 Rotational invariance now gives every remaining large zero as
2070 $\omega^k\tau_m$.  Distinct rays and the disjoint disks give uniqueness
2071 of $k$ and $m$.  Taking $\gamma=\gamma_N$ proves the zero location in
2072 the statement.
2073 
2074 Differentiate the normalized expression.  At $z=\tau_m$,
2075 \[
2076 G'(\tau_m)
2077 =
2078 2is_Ne^{2is_N\tau_m}+R_N'(\tau_m)
2079 =
2080 -2is_N+O(e^{-\gamma_N\tau_m}).
2081 \]
2082 Since $G(\tau_m)=0$,
2083 \[
2084 \Dfun_N'(\tau_m)
2085 =
2086 \frac1N e^{e^{-ia_N}\tau_m}G'(\tau_m).
2087 \]
2088 Taking absolute values gives the derivative asymptotic.  Differentiating $\Dfun_N(\omega z)=\Dfun_N(z)$ gives the rotation formula.
2089 \end{proof}
2090 
2091 \begin{theorem}[Leading displacement term and derivative sign]\label{thm:first_zero_correction}
2092 With the notation of \cref{lem:two_term_dominance,thm:large_zero_strings},
2093 \[
2094 \tau_m
2095 =
2096 t_m+
2097 \frac{R_N(t_m)}{2is_N}
2098 +
2099 O(e^{-2\gamma_Nt_m}).
2100 \]
2101 The quotient $R_N(t_m)/(2is_N)$ is real.  One also has
2102 \[
2103 \Dfun_N'(\tau_m)
2104 =
2105 (-1)^{m+1}
2106 \frac{2s_N}{N}
2107 e^{c_N\tau_m}
2108 \left(1+O(e^{-\gamma_N\tau_m})\right).
2109 \]
2110 \end{theorem}
2111 
2112 \begin{proof}
2113 Put $\delta_m=\tau_m-t_m$.  The large zero theorem gives $\delta_m=O(e^{-\gamma_Nt_m})$.  Since $e^{2is_Nt_m}=-1$,
2114 \[
2115 1+e^{2is_N(t_m+\delta_m)}
2116 =
2117 -2is_N\delta_m+O(\delta_m^2).
2118 \]
2119 The remainder estimate through the first derivative gives
2120 \[
2121 R_N(t_m+\delta_m)
2122 =
2123 R_N(t_m)+O(e^{-\gamma_Nt_m}|\delta_m|).
2124 \]
2125 Substitution in the zero equation gives the displacement formula.
2126 
2127 At the real point $t_m$, the normalized value
2128 \[
2129 N e^{-e^{-ia_N}t_m}\Dfun_N(t_m)
2130 \]
2131 is purely imaginary.  The two dominant terms cancel there, so $R_N(t_m)$ is purely imaginary and the displayed correction is real.
2132 
2133 For the derivative, use the formula in the proof of \cref{thm:large_zero_strings} and
2134 \[
2135 e^{-is_N\tau_m}
2136 =
2137 (-1)^m(-i)
2138 \left(1+O(e^{-\gamma_N\tau_m})\right).
2139 \]
2140 The stated real sign follows.
2141 \end{proof}
2142 
2143 \begin{corollary}[Leading correction for $N=3$]\label{cor:N3_zero_correction}
2144 For $N=3$,
2145 \[
2146 t_m=\frac{2\pi}{\sqrt3}\left(m+\frac12\right)
2147 \]
2148 and
2149 \[
2150 \tau_m
2151 =
2152 t_m+
2153 \frac{(-1)^m}{\sqrt3}e^{-3t_m/2}
2154 +
2155 O(e^{-3t_m}).
2156 \]
2157 \end{corollary}
2158 
2159 \begin{proof}
2160 The only nondominant exponent is $-1$.  Hence
2161 \[
2162 R_3(t)
2163 =
2164 \exp\left(\left(-\frac32+\frac{\sqrt3}{2}i\right)t\right).
2165 \]
2166 At $t=t_m$, its phase is $i(-1)^m$.  Insert this value into \cref{thm:first_zero_correction}.
2167 \end{proof}
2168 
2169 \begin{corollary}[The case $N=2$]\label{cor:case_two_zeros}
2170 For $N=2$,
2171 \[
2172 \Dfun_2(z)=\cos z.
2173 \]
2174 Its zeros are
2175 \[
2176 \pm\left(m+\frac12\right)\pi,
2177 \qquad m\geq0,
2178 \]
2179 and all are simple.  They agree with the formula $t_m=A_2(m+1/2)$.
2180 \end{corollary}
2181 
2182 \begin{proof}
2183 This is the standard zero set of the cosine function.
2184 \end{proof}
2185 
2186 \begin{corollary}[Eventual simplicity and exact rays]\label{cor:eventual_simplicity}
2187 For every $N\geq2$, all but finitely many zeros of $\Dfun_N$ are simple.  Every sufficiently large zero lies on one of the rays
2188 \[
2189 \omega^k(0,\infty),
2190 \qquad 0\leq k<N.
2191 \]
2192 \end{corollary}
2193 
2194 \begin{proof}
2195 Use \cref{thm:large_zero_strings} for $N\geq3$ and \cref{cor:case_two_zeros} for $N=2$.
2196 \end{proof}
2197 
2198 
2199 \begin{example}[Exact fourth-order strings]\label{ex:fourth_order_zeros}
2200 For $N=4$,
2201 \[
2202 \Dfun_4(z)
2203 =
2204 \cos\left(\frac z{\sqrt2}\right)
2205 \cosh\left(\frac z{\sqrt2}\right).
2206 \]
2207 The positive zeros are exactly
2208 \[
2209 \tau_m
2210 =
2211 \sqrt2\pi\left(m+\frac12\right)
2212 =
2213 A_4\left(m+\frac12\right).
2214 \]
2215 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.
2216 \end{example}
2217 
2218 \begin{corollary}[Reduced zeros]\label{cor:reduced_zeros}
2219 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
2220 \[
2221 \lambda_m=\tau_m^N
2222 \]
2223 for all sufficiently large $m$.  If $N\geq3$, the notation may be chosen
2224 as in \cref{thm:large_zero_strings}, and
2225 \[
2226 \lambda_m
2227 =
2228  t_m^N
2229 +O\left(t_m^{N-1}e^{-\gamma t_m}\right).
2230 \]
2231 For $N=2$, take $\tau_m=t_m=\pi(m+1/2)$ for $m\geq0$. Then
2232 $\lambda_m=t_m^2$ exactly.
2233 \end{corollary}
2234 
2235 \begin{proof}
2236 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
2237 \[
2238 \Dfun_N'(z)=Nz^{N-1}\Delta_N'(z^N)
2239 \]
2240 and $z\ne0$.  The asymptotic follows by expanding $\tau_m^N$ about $t_m$.
2241 \end{proof}
2242 
2243 \begin{theorem}[Zero counting asymptotics]\label{thm:zero_counting}
2244 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 $R\to\infty$,
2245 \[
2246 n_{\Dfun_N}(R)
2247 =
2248 \frac{N\sin(\pi/N)}{\pi}R+O(1)
2249 \]
2250 and
2251 \[
2252 n_{\Delta_N}(R)
2253 =
2254 \frac{\sin(\pi/N)}{\pi}R^{1/N}+O(1).
2255 \]
2256 \end{theorem}
2257 
2258 \begin{proof}
2259 Each of the $N$ rays contains one large zero near
2260 \[
2261  t_m=\frac\pi{\sin(\pi/N)}\left(m+\frac12\right).
2262 \]
2263 Since $\tau_m-t_m=o(1)$ and the model spacing is the positive constant
2264 $A_N$, replacing $\tau_m\leq R$ by $t_m\leq R$ changes the count by at
2265 most a bounded amount.
2266 The number of such indices with $t_m\leq R$ is
2267 \[
2268 \frac{\sin(\pi/N)}\pi R+O(1).
2269 \]
2270 Multiplication by $N$ and addition of the finite exceptional set give the first formula.  The second follows from
2271 \[
2272 n_{\Dfun_N}(R^{1/N})=Nn_{\Delta_N}(R).
2273 \]
2274 \end{proof}
2275 
2276 
2277 \begin{corollary}[Zero exponents]\label{cor:exponent_of_convergence}
2278 The exponent of convergence of the nonzero zeros of $\Dfun_N$ is one.  The exponent of convergence of the zeros of $\Delta_N$ is $1/N$.
2279 \end{corollary}
2280 
2281 \begin{proof}
2282 The asymptotic counting formulas in \cref{thm:zero_counting} give the two exponents by partial summation.
2283 \end{proof}
2284 
2285 \clearpage\part{Products, moments, and zero zeta functions}
2286 \section{Hadamard products and zero power sums}\label{sec:hadamard}
2287 
2288 \begin{keyidea}
2289 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.
2290 \end{keyidea}
2291 \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}
2292 
2293 
2294 The reduction from $z$ to $w=z^N$ removes the rotational repetition of the zeros.
2295 
2296 \begin{theorem}[Genus zero product]\label{thm:genus_zero_product}
2297 For $N\geq2$, let $\Lambda_N$ be the nonzero zero multiset of $\Delta_N$.  Then
2298 \[
2299 \Delta_N(w)
2300 =
2301 \prod_{\lambda\in\Lambda_N}
2302 \left(1-\frac w\lambda\right),
2303 \]
2304 where the product converges locally uniformly.  Consequently,
2305 \[
2306 \Dfun_N(z)
2307 =
2308 \prod_{\lambda\in\Lambda_N}
2309 \left(1-\frac{z^N}{\lambda}\right).
2310 \]
2311 \end{theorem}
2312 
2313 \begin{proof}
2314 By \cref{prop:orders}, the order of $\Delta_N$ is $1/N<1$.  By
2315 \cref{cor:reduced_zeros}, it has infinitely many zeros, and
2316 \cref{cor:exponent_of_convergence} gives
2317 \[
2318 \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty.
2319 \]
2320 Hadamard factorization therefore gives a canonical product of genus zero.
2321 The only possible exponential factor is a nonzero constant because the order
2322 is below one.  Since $\Delta_N(0)=1$, that constant is one.  Substitution of
2323 $w=z^N$ gives the second formula.
2324 \end{proof}
2325 
2326 
2327 \begin{theorem}[Diagonal Fredholm determinant representation]\label{thm:trace_class_model}
2328 Let $I_N$ be a multiplicity-sensitive countable index set for $\Lambda_N$
2329 and put $\mathcal H_N=\ell^2(I_N)$.  On its standard orthonormal basis,
2330 define the diagonal operator
2331 \[
2332 K_Ne_\lambda=\lambda^{-1}e_\lambda.
2333 \]
2334 Then $K_N$ is trace class and
2335 \[
2336 \Delta_N(w)=\Fdet(I-wK_N).
2337 \]
2338 For every $m\geq1$,
2339 \[
2340 \kappa_{N,m}=\tr(K_N^m).
2341 \]
2342 \end{theorem}
2343 
2344 \begin{proof}
2345 By \cref{cor:exponent_of_convergence},
2346 \[
2347 \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty.
2348 \]
2349 Thus the diagonal operator $K_N$ is trace class.  Its Fredholm determinant is the locally uniform product
2350 \[
2351 \Fdet(I-wK_N)
2352 =
2353 \prod_{\lambda\in\Lambda_N}
2354 \left(1-\frac w\lambda\right),
2355 \]
2356 which equals $\Delta_N(w)$ by \cref{thm:genus_zero_product}.
2357 
2358 Taking the logarithm gives
2359 \[
2360 -\log\Delta_N(w)
2361 =
2362 \sum_{m\geq1}\frac{w^m}{m}\tr(K_N^m)
2363 \]
2364 near the origin.  Comparison with the definition of $\kappa_{N,m}$ proves the trace identity.
2365 \end{proof}
2366 
2367 This is the standard diagonal encoding of this family's genus zero product.
2368 It is not a general trace-class resolvent or Feshbach theorem.
2369 
2370 \begin{corollary}[Complete symmetric function form]\label{cor:complete_symmetric_form}
2371 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
2372 \[
2373 \frac{P_{N,n}(r)}{(Nn)!}=h_n.
2374 \]
2375 For arbitrary complex $r$, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by $r$.
2376 \end{corollary}
2377 
2378 \begin{proof}
2379 The generating function for the complete homogeneous symmetric functions is
2380 \[
2381 \prod_{\lambda\in\Lambda_N}
2382 \left(1-\frac w\lambda\right)^{-r}
2383 =
2384 \Delta_N(w)^{-r}.
2385 \]
2386 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}.
2387 \end{proof}
2388 
2389 
2390 \begin{proposition}[Products for the normalized sections]\label{prop:normalized_section_product}
2391 For every $0\leq j<N$, the entire function
2392 \[
2393 \Phi_{N,j}(w)
2394 =
2395 j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!}
2396 \]
2397 has order $1/N$.  If $N\geq2$ and $\Lambda_{N,j}$ is its nonzero zero multiset, then
2398 \[
2399 \Phi_{N,j}(w)
2400 =
2401 \prod_{\mu\in\Lambda_{N,j}}
2402 \left(1-\frac w\mu\right).
2403 \]
2404 For $m\geq1$, put
2405 \[
2406 \sigma_{N,j,m}
2407 =
2408 \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}.
2409 \]
2410 Then
2411 \[
2412 C_{N,j,n}(r)
2413 =
2414 \frac{(Nn)!}{n!}
2415 B_n\bigl(
2416  r\sigma_{N,j,1},
2417  r1!\sigma_{N,j,2},
2418  \ldots,
2419  r(n-1)!\sigma_{N,j,n}
2420 \bigr).
2421 \]
2422 \end{proposition}
2423 
2424 \begin{proof}
2425 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}.
2426 
2427 Taking the logarithm gives
2428 \[
2429 -\log\Phi_{N,j}(w)
2430 =
2431 \sum_{m\geq1}\sigma_{N,j,m}\frac{w^m}{m}.
2432 \]
2433 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.
2434 \end{proof}
2435 
2436 \begin{definition}\label{def:zero_power_sums}
2437 For $m\geq1$, define
2438 \[
2439 \sigma_{N,m}
2440 =
2441 \sum_{\lambda\in\Lambda_N}\lambda^{-m},
2442 \]
2443 with multiplicities.
2444 \end{definition}
2445 
2446 The series converges absolutely because the exponent of convergence of $\Lambda_N$ is $1/N<1$.
2447 
2448 \begin{theorem}[Newton sums for the reduced zeros]\label{thm:newton_sums}
2449 For $|w|$ below the least zero modulus, with the analytic logarithm
2450 normalized to zero at the origin,
2451 \[
2452 -\log\Delta_N(w)
2453 =
2454 \sum_{m\geq1}\sigma_{N,m}\frac{w^m}{m}.
2455 \]
2456 Thus
2457 \[
2458 \kappa_{N,m}=\sigma_{N,m}.
2459 \]
2460 If
2461 \[
2462 a_m=\frac{(-1)^m}{(Nm)!},
2463 \qquad a_0=1,
2464 \]
2465 then
2466 \[
2467 \sigma_{N,m}
2468 =
2469 -ma_m-
2470 \sum_{k=1}^{m-1}a_k\sigma_{N,m-k}.
2471 \]
2472 \end{theorem}
2473 
2474 \begin{proof}
2475 Take the logarithm of the product in \cref{thm:genus_zero_product}.  Absolute convergence in a disk below the first zero permits termwise expansion
2476 \[
2477 -\log\left(1-\frac w\lambda\right)
2478 =
2479 \sum_{m\geq1}\frac{w^m}{m\lambda^m}.
2480 \]
2481 This gives the logarithmic formula and identifies $\kappa_{N,m}$ with $\sigma_{N,m}$.
2482 
2483 Differentiate the product identity in logarithmic form to obtain
2484 \[
2485 \Delta_N'(w)
2486 =
2487 -\Delta_N(w)
2488 \sum_{m\geq1}\sigma_{N,m}w^{m-1}.
2489 \]
2490 Comparison of the coefficient of $w^{m-1}$ gives
2491 \[
2492 ma_m
2493 =
2494 -\sum_{k=0}^{m-1}a_k\sigma_{N,m-k}.
2495 \]
2496 Solving for $\sigma_{N,m}$ gives the recurrence.
2497 \end{proof}
2498 
2499 
2500 
2501 \begin{corollary}[Convergent Mittag--Leffler partial-fraction form]\label{cor:mittag_leffler_tangent}
2502 Away from the zeros of $\Dfun_N$,
2503 \[
2504 \Tfun_N(z)
2505 =
2506 Nz^{N-1}
2507 \sum_{\lambda\in\Lambda_N}
2508 \frac1{\lambda-z^N}.
2509 \]
2510 The series converges absolutely and locally uniformly on compact sets
2511 away from the poles.  Thus no regularizing terms are required.
2512 \end{corollary}
2513 
2514 \begin{proof}
2515 Logarithmic differentiation of the genus zero product gives
2516 \[
2517 -\frac{\Delta_N'(w)}{\Delta_N(w)}
2518 =
2519 \sum_{\lambda\in\Lambda_N}\frac1{\lambda-w}.
2520 \]
2521 The series is locally uniform because $\sum_{\lambda}|\lambda|^{-1}<\infty$.  Substitute $w=z^N$ and use the chain rule.
2522 \end{proof}
2523 
2524 \begin{proposition}[Theta Bell polynomial formula]\label{prop:theta_bell}
2525 Put
2526 \[
2527 \widetilde{\sigma}_{N,m}=(-1)^m\sigma_{N,m}.
2528 \]
2529 Then
2530 \[
2531 \Thetafun_{N,n}(u)
2532 =
2533 \frac{(Nn)!}{n!}
2534 B_n\bigl(
2535  \widetilde{\sigma}_{N,1}(u^N+u),
2536  1!\widetilde{\sigma}_{N,2}(u^{2N}+u),
2537  \ldots,
2538  (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u)
2539 \bigr).
2540 \]
2541 \end{proposition}
2542 
2543 \begin{proof}
2544 The reduced unsigned denominator is
2545 \[
2546 \Psi_N(w)=\sum_{n\geq0}\frac{w^n}{(Nn)!}=\Delta_N(-w).
2547 \]
2548 By \cref{thm:newton_sums},
2549 \[
2550 -\log\Psi_N(w)
2551 =
2552 \sum_{m\geq1}\widetilde{\sigma}_{N,m}\frac{w^m}{m}.
2553 \]
2554 With $w=z^N$, the logarithm of the theta generating function is
2555 \[
2556 \sum_{m\geq1}
2557 \widetilde{\sigma}_{N,m}(u^{Nm}+u)\frac{w^m}{m}.
2558 \]
2559 The complete Bell polynomial formula follows.
2560 \end{proof}
2561 
2562 \begin{corollary}[Tangent coefficients as zero sums]\label{cor:tangent_zero_sums}
2563 For every $n\geq0$,
2564 \[
2565 T_N(n)
2566 =
2567 N(Nn+N-1)!\sigma_{N,n+1}.
2568 \]
2569 \end{corollary}
2570 
2571 \begin{proof}
2572 Combine \cref{prop:tangent_kappa,thm:newton_sums}.
2573 \end{proof}
2574 
2575 \begin{corollary}[Reciprocal coefficients from zero data]\label{cor:reciprocal_zero_data}
2576 For every $n\geq0$,
2577 \[
2578 P_{N,n}(r)
2579 =
2580 \frac{(Nn)!}{n!}
2581 B_n\bigl(
2582  r\sigma_{N,1},
2583  r1!\sigma_{N,2},
2584  \ldots,
2585  r(n-1)!\sigma_{N,n}
2586 \bigr).
2587 \]
2588 \end{corollary}
2589 
2590 \begin{proof}
2591 Use \cref{thm:reciprocal_bell,thm:newton_sums}.
2592 \end{proof}
2593 
2594 \begin{example}[The cosine orbit sums]\label{ex:cosine_orbit_sums}
2595 For $N=2$,
2596 \[
2597 \Delta_2(w)=\cos\sqrt w.
2598 \]
2599 Its zeros are
2600 \[
2601 \lambda_m=\pi^2\left(m+\frac12\right)^2,
2602 \qquad m\geq0.
2603 \]
2604 Hence
2605 \[
2606 \sigma_{2,q}
2607 =
2608 \frac{2^{2q}-1}{\pi^{2q}}\zeta(2q).
2609 \]
2610 The identity in \cref{cor:tangent_zero_sums} recovers the classical tangent coefficients from these zero power sums.
2611 \end{example}
2612 
2613 \section{Coefficient asymptotics from the nearest zeros}\label{sec:coefficient_asymptotics}
2614 
2615 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 $r$.
2616 
2617 \begin{theorem}[Nearest-pole coefficient expansion]\label{thm:nearest_circle_expansion}
2618 Let $r\in\ZZ_{>0}$.  Suppose the zeros of $\Dfun_N$ of least modulus
2619 $R$ are simple and form the finite set $\Zcal_R$.  Suppose the next
2620 zero modulus is $R_2>R$.  At each $\rho\in\Zcal_R$, write the principal
2621 part
2622 \[
2623 \Dfun_N(z)^{-r}
2624 =
2625 \sum_{\ell=1}^{r}
2626  a_{\rho,\ell}
2627 \left(1-\frac z\rho\right)^{-\ell}
2628 +
2629 O(1).
2630 \]
2631 Then, for every $R_*$ with $R<R_*<R_2$ and $m=Nn$, as $n\to\infty$,
2632 \[
2633 P_{N,n}(r)
2634 =
2635  m!
2636 \sum_{\rho\in\Zcal_R}
2637 \sum_{\ell=1}^{r}
2638  a_{\rho,\ell}
2639 \binom{m+\ell-1}{\ell-1}
2640 \rho^{-m}
2641 +
2642 O_{R_*}(m!R_*^{-m}).
2643 \]
2644 The leading Laurent coefficient is
2645 \[
2646 a_{\rho,r}
2647 =
2648 (-\rho\Dfun_N'(\rho))^{-r}.
2649 \]
2650 \end{theorem}
2651 
2652 \begin{proof}
2653 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
2654 \[
2655 \left(1-\frac z\rho\right)^{-\ell}
2656 \]
2657 is
2658 \[
2659 \binom{m+\ell-1}{\ell-1}\rho^{-m}.
2660 \]
2661 Multiplication by $m!$ converts the ordinary coefficient to $P_{N,n}(r)$.
2662 
2663 Near a simple zero,
2664 \[
2665 \Dfun_N(z)
2666 =
2667 -\rho\Dfun_N'(\rho)
2668 \left(1-\frac z\rho\right)
2669 +O\left(\left(1-\frac z\rho\right)^2\right).
2670 \]
2671 Raising to the power $-r$ gives the formula for $a_{\rho,r}$.
2672 \end{proof}
2673 
2674 \begin{corollary}[Leading dominant zero term]\label{cor:leading_dominant_zero}
2675 Under the hypotheses of \cref{thm:nearest_circle_expansion},
2676 \[
2677 \begin{aligned}
2678 P_{N,n}(r)
2679 ={}&m!\binom{m+r-1}{r-1}
2680 \sum_{\rho\in\Zcal_R}
2681 \frac{\rho^{-m}}{(-\rho\Dfun_N'(\rho))^r}
2682 \\
2683 &+O\left(m!m^{r-2}R^{-m}\right)
2684 +O_{R_*}\left(m!R_*^{-m}\right).
2685 \end{aligned}
2686 \]
2687 For $r=1$, the first error term is absent.
2688 \end{corollary}
2689 
2690 \begin{proof}
2691 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.
2692 \end{proof}
2693 
2694 
2695 \begin{corollary}[Orbit reduced nearest circle term]\label{cor:orbit_reduced_asymptotic}
2696 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 $m=Nn$,
2697 \[
2698 \begin{aligned}
2699 P_{N,n}(r)
2700 ={}&N m!\binom{m+r-1}{r-1}
2701 \sum_{\tau\in\Zcal_R^{\mathrm{orb}}}
2702 \frac{\tau^{-m}}{(-\tau\Dfun_N'(\tau))^r}
2703 \\
2704 &+O\left(m!m^{r-2}R^{-m}\right)
2705 +O_{R_*}\left(m!R_*^{-m}\right).
2706 \end{aligned}
2707 \]
2708 For $r=1$, the first error term is absent.
2709 \end{corollary}
2710 
2711 \begin{proof}
2712 If $\rho=\omega^k\tau$, then
2713 \[
2714 \Dfun_N'(\rho)=\omega^{-k}\Dfun_N'(\tau).
2715 \]
2716 Therefore
2717 \[
2718 -\rho\Dfun_N'(\rho)
2719 =-\tau\Dfun_N'(\tau).
2720 \]
2721 Since $m=Nn$, one also has $\rho^{-m}=\tau^{-m}$.  Every orbit therefore contributes $N$ equal terms in \cref{cor:leading_dominant_zero}.
2722 \end{proof}
2723 
2724 \begin{corollary}[Secant asymptotic]\label{cor:secant_asymptotic}
2725 For $N=2$ and $r=1$,
2726 \[
2727 P_{2,n}(1)
2728 \sim
2729 2(2n)!\left(\frac2\pi\right)^{2n+1}.
2730 \]
2731 \end{corollary}
2732 
2733 \begin{proof}
2734 The nearest zeros of $\cos z$ are $\rho=\pm\pi/2$, both simple.  Their contributions in \cref{cor:leading_dominant_zero} are equal.
2735 \end{proof}
2736 
2737 \section{Zero zeta functions}\label{sec:zero_zeta}
2738 
2739 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.
2740 
2741 \begin{definition}\label{def:absolute_zero_zetas}
2742 Define
2743 \[
2744 \mathfrak Z_N^{\mathrm{abs}}(s)
2745 =
2746 \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
2747 \]
2748 and
2749 \[
2750 \mathfrak Z_N^{\mathrm{red}}(s)
2751 =
2752 \sum_{\lambda\in\Lambda_N}|\lambda|^{-s},
2753 \]
2754 with multiplicities.  Each term means $e^{-s\log|\rho|}$ or $e^{-s\log|\lambda|}$.
2755 \end{definition}
2756 
2757 \begin{theorem}[Abscissae of absolute convergence]\label{thm:zeta_convergence}
2758 The series $\mathfrak Z_N^{\mathrm{abs}}$ converges absolutely and locally uniformly when
2759 \[
2760 \Re s>1.
2761 \]
2762 It fails to converge absolutely when $\Re s\leq1$.  The series $\mathfrak Z_N^{\mathrm{red}}$ converges absolutely and locally uniformly when
2763 \[
2764 \Re s>\frac1N.
2765 \]
2766 It fails to converge absolutely when $\Re s\leq1/N$.  Both functions are holomorphic on their open half-planes of absolute convergence.
2767 \end{theorem}
2768 
2769 \begin{proof}
2770 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.
2771 \end{proof}
2772 
2773 \begin{theorem}[Meromorphic continuation of the symmetry-orbit zero zeta]\label{thm:zeta_continuation}
2774 Put
2775 \[
2776 A_N=\frac\pi{\sin(\pi/N)}.
2777 \]
2778 There is an entire function $Q_N^{\mathrm{red}}$ such that
2779 \[
2780 \mathfrak Z_N^{\mathrm{red}}(s)
2781 =
2782 A_N^{-Ns}\zeta\left(Ns,\frac12\right)+Q_N^{\mathrm{red}}(s)
2783 \]
2784 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
2785 \[
2786 \frac{\sin(\pi/N)}{N\pi}.
2787 \]
2788 
2789 There is an entire function $Q_N^{\mathrm{abs}}$ such that, initially on
2790 $\Re s>1$,
2791 \[
2792 \mathfrak Z_N^{\mathrm{abs}}(s)
2793 =
2794 N A_N^{-s}\zeta\left(s,\frac12\right)+Q_N^{\mathrm{abs}}(s).
2795 \]
2796 Thus $\mathfrak Z_N^{\mathrm{abs}}$ continues meromorphically to $\CC$ with one simple pole at $s=1$, of residue
2797 \[
2798 \frac{N\sin(\pi/N)}\pi.
2799 \]
2800 \end{theorem}
2801 
2802 \begin{proof}
2803 For $N=2$, \cref{cor:case_two_zeros} and
2804 $\Dfun_2(z)=\Delta_2(z^2)$ give
2805 \[
2806 \lambda_m=\pi^2\left(m+\frac12\right)^2,
2807 \qquad m\geq0.
2808 \]
2809 Thus the first formula holds with $Q_2^{\mathrm{red}}=0$.
2810 
2811 For $N\geq3$, \cref{cor:reduced_zeros} writes all but finitely many reduced zeros as
2812 \[
2813 \lambda_m
2814 =
2815 \left(A_N\left(m+\frac12\right)\right)^N(1+\varepsilon_m)
2816 \]
2817 where $\lambda_m>0$, $1+\varepsilon_m>0$, and
2818 \[
2819 \varepsilon_m
2820 =
2821 O\left(t_m^{-1}e^{-\gamma t_m}\right).
2822 \]
2823 On every compact set of $s$ values,
2824 \[
2825 (1+\varepsilon_m)^{-s}-1
2826 =
2827 O(|\varepsilon_m|)
2828 \]
2829 uniformly for large $m$.  The difference
2830 \[
2831 \lambda_m^{-s}
2832 -
2833 A_N^{-Ns}\left(m+\frac12\right)^{-Ns}
2834 \]
2835 therefore has a normally convergent sum on every compact subset of
2836 $\CC$.  Indeed, the model power has at most polynomial growth in $m$
2837 on a compact set, while $|\varepsilon_m|$ decays exponentially.  The
2838 finitely many exceptional terms $|\lambda|^{-s}$ and the finitely many
2839 omitted model terms are entire functions of $s$.  Their addition to the
2840 normally convergent tail defines $Q_N^{\mathrm{red}}$ and proves the
2841 first formula. Thus that formula is established for every $N\geq2$.
2842 
2843 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.
2844 
2845 For every $N\geq2$, each nonzero zero $\lambda$ of $\Delta_N$ gives $N$ zeros of $\Dfun_N$, each of modulus $|\lambda|^{1/N}$.  Hence
2846 \[
2847 \mathfrak Z_N^{\mathrm{abs}}(s)
2848 =
2849 N\mathfrak Z_N^{\mathrm{red}}(s/N).
2850 \]
2851 Substitution gives the second formula with
2852 \[
2853 Q_N^{\mathrm{abs}}(s)=NQ_N^{\mathrm{red}}(s/N),
2854 \]
2855 which is entire.
2856 \end{proof}
2857 
2858 \begin{corollary}[Exact zeta models in orders two and four]\label{cor:exact_zeta_models}
2859 For $N=2$,
2860 \[
2861 \mathfrak Z_2^{\mathrm{red}}(s)
2862 =
2863 \pi^{-2s}\zeta\left(2s,\frac12\right)
2864 \]
2865 and
2866 \[
2867 \mathfrak Z_2^{\mathrm{abs}}(s)
2868 =
2869 2\pi^{-s}\zeta\left(s,\frac12\right).
2870 \]
2871 For $N=4$, put $A_4=\sqrt2\pi$.  Then
2872 \[
2873 \mathfrak Z_4^{\mathrm{red}}(s)
2874 =
2875 A_4^{-4s}\zeta\left(4s,\frac12\right)
2876 \]
2877 and
2878 \[
2879 \mathfrak Z_4^{\mathrm{abs}}(s)
2880 =
2881 4A_4^{-s}\zeta\left(s,\frac12\right).
2882 \]
2883 In these two cases the entire correction terms in \cref{thm:zeta_continuation} vanish identically.
2884 \end{corollary}
2885 
2886 \begin{proof}
2887 For $N=2$, the reduced zeros are
2888 \[
2889 \pi^2\left(m+\frac12\right)^2,
2890 \qquad m\geq0.
2891 \]
2892 For $N=4$, \cref{ex:fourth_order_zeros} gives the reduced zeros
2893 \[
2894 A_4^4\left(m+\frac12\right)^4,
2895 \qquad m\geq0.
2896 \]
2897 Summing their negative powers proves the reduced formulas.  The absolute formulas follow from
2898 \[
2899 \mathfrak Z_N^{\mathrm{abs}}(s)
2900 =
2901 N\mathfrak Z_N^{\mathrm{red}}(s/N).
2902 \]
2903 \end{proof}
2904 
2905 \begin{definition}\label{def:branch_orbit_zeta}
2906 Choose a logarithm for each member of the finite exceptional part of $\Lambda_N$.  Use the real logarithm for the eventual positive sequence.  Define
2907 \[
2908 \mathfrak Z_N^{\mathrm{orb}}(s)
2909 =
2910 \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
2911 \]
2912 This series is initially defined on $\Re s>1/N$.
2913 \end{definition}
2914 
2915 \begin{corollary}[Branch-dependent symmetry-orbit continuation]\label{cor:branch_grouped_continuation}
2916 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.
2917 \end{corollary}
2918 
2919 \begin{proof}
2920 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.
2921 \end{proof}
2922 
2923 \begin{definition}\label{def:derivative_weighted_zeta}
2924 Let $q$ be a positive integer.  Choose logarithms of the simple zeros of $\Dfun_N$ with arguments in a fixed bounded interval.  Define
2925 \[
2926 \mathfrak Z_{N,q}^{\mathrm{der}}(s)
2927 =
2928 \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}}
2929 \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
2930 \]
2931 \end{definition}
2932 
2933 \begin{theorem}[Derivative-weighted convergence]\label{thm:derivative_weighted_convergence}
2934 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.
2935 
2936 For $N=2$, the series converges absolutely and locally uniformly on
2937 $\Re s>1$, where it defines a holomorphic function.
2938 \end{theorem}
2939 
2940 \begin{proof}
2941 For $N\geq3$, \cref{thm:large_zero_strings} gives
2942 \[
2943 |\Dfun_N'(\rho)|^{-1}
2944 \leq
2945 C e^{-c_N|\rho|}
2946 \]
2947 for every sufficiently large simple zero.  Let $L$ be a compact subset
2948 of the $s$ plane.  Because the chosen arguments of the zeros lie in a
2949 fixed bounded interval, there are constants $C_L$ and $M_L$ such that
2950 \[
2951 \sup_{s\in L}|e^{-s\Log\rho}|
2952 \leq C_L|\rho|^{M_L}
2953 \]
2954 for every sufficiently large zero.  Hence the absolute value of the
2955 corresponding summand is at most
2956 \[
2957 C_L'|\rho|^{M_L}e^{-q c_N|\rho|}.
2958 \]
2959 The linear zero counting estimate makes this majorant summable.  The
2960 Weierstrass test gives absolute local uniform convergence of the
2961 series of entire summands, so its sum is entire.  The finite exceptional
2962 set causes no difficulty.
2963 
2964 For $N=2$, the derivative of the cosine has modulus one at every zero.  The assertion then follows from the linear zero spacing.
2965 \end{proof}
2966 
2967 \begin{remark}\label{rem:multiple_zeros}
2968 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.
2969 \end{remark}
2970 
2971 \clearpage\part{Scope and formal evidence}
2972 \section{Scope boundaries and further questions}\label{sec:further_questions}
2973 
2974 The large-zero theorem settles eventual simplicity but does not classify the
2975 finite exceptional set. A remaining question is whether every zero of
2976 $\Dfun_N$ is simple and every reduced zero of $\Delta_N$ is positive. Both
2977 statements hold for $N=2$ and $N=4$. The argument here proves them only outside
2978 a compact set for general $N$.
2979 
2980 The leading correction in \cref{thm:first_zero_correction} may admit iteration
2981 to a full exponential expansion for $\tau_m$. Such an expansion is an open
2982 problem, not a result of this chapter. It could refine the entire correction in
2983 the zero-zeta comparison and lead to explicit special values.
2984 
2985 The symmetry-orbit zero zeta has a meromorphic continuation with one pole. No
2986 functional equation is obtained. A branch dependent sum over the $N$ roots of
2987 each reduced zero must specify logarithms on the rays.
2988 
2989 The coefficient families in
2990 \cref{sec:reciprocal_powers,sec:residue_section,sec:theta} have exact formal
2991 parameter-addition laws. Their arithmetic behavior, including odd-prime
2992 congruences, dyadic weighting, and $p$-adic or Amice--Mahler interpolation,
2993 belongs to Chapter IX. General bounded-operator semigroup, resolvent, and
2994 Feshbach theory belongs to Chapter VI. Chapter III covers general nonlinear
2995 operation trees, while abstract completion and occupancy questions belong to
2996 Chapter X. None of those theories is inferred from the scalar complex-analytic
2997 results proved here.
2998 
2999 \section{Formalization ledger}\label{sec:formalization_ledger}
3000 
3001 The recorded formal verification used Lean 4 v4.34.0-rc2 and Mathlib revision
3002 \leanname{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}
3003 \cite{deMouraUllrich2021,mathlib2020}. Its 151 theorem-map rows resolve to
3004 1164 unique declarations in the namespace \leanname{Komorebi.PaperIV}:
3005 953 theorem declarations and 211 constructions. The recorded namespace audit scanned
3006 2234 declarations, including 1999 theorem declarations. The recorded build with warnings treated as errors
3007 and supplied kernel audit allowed only
3008 \leanname{propext}, \leanname{Classical.choice}, and
3009 \leanname{Quot.sound} as standard transitive axioms.
3010 
3011 \begingroup
3012 \small
3013 \begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}p{0.70\textwidth}@{}}
3014 \toprule
3015 Manuscript records & Compiled Chapter IV scope \\
3016 \midrule
3017 \endfirsthead
3018 \toprule
3019 Manuscript records & Compiled Chapter IV scope (continued) \\
3020 \midrule
3021 \endhead
3022 3.1--3.3 & Actual holomorphic root filters, convergent coefficient
3023 selection, orthogonal resolution, the exact range--covariance characterization,
3024 and the complete pointwise cyclic product law. The formal versions hold as well. \\
3025 4.1--4.5 and 5.1--5.8 & Entire sections, finite exponential formulas,
3026 derivative cycles, both Wronskians, and unique entire ODE basis expansions.
3027 Both pointwise addition laws and denominator symmetries are checked.
3028 The exact ordinary hypergeometric formula follows from the omitted-parameter
3029 list, Gamma normalization, and Mathlib's everywhere convergent series. \\
3030 6.1--6.6 & Quotient-algebra power bases and actual exponential-series
3031 convergence in the finite-dimensional coefficient topology. Both pointwise
3032 section-coordinate identities. Signed-shift powers, all matrix-exponential
3033 entries, group laws and trace are checked. The characteristic polynomial
3034 $X^N+1$, its distinct roots, exact spectrum and determinant one are proved. \\
3035 7.1--7.8 & Named reciprocal families, exact parameter recurrences, Bell
3036 formula, parameter degree and leading coefficient, and polynomial divisibility.
3037 Parameter addition holds with any single fixed holomorphic logarithm on the
3038 stated domain. Newton's recurrence gives the literal Hessenberg determinant.
3039 All three compact reciprocal-polynomial formulas are checked. \\
3040 8.1--8.5 & The full tangent coefficient series has the exact kappa and
3041 factorial normalization, including $N=1$. Reciprocal differentiation holds on
3042 every stated fixed logarithm domain. The actual higher Riccati identity holds
3043 at every point where the denominator is nonzero. \\
3044 9.1--9.10 & Arbitrary-parameter residue reciprocals and their analytic germs,
3045 polynomial degree and leading term, sparse Appell laws, joint convolution,
3046 monicity and covariance. The Hasse and derivative-truncation encodings are
3047 proved equal. The named Appell polynomial satisfies both the forward action
3048 and the exact $\Phi_{N,j}(\partial_x^N)^r$ inverse formula. \\
3049 10.1--10.4 & Exact formal theta reconstruction, polynomial lift and degree,
3050 literal divisibility by $u$ and, for even $N$, by $u+1$, and the theta Newton
3051 recurrence with its first coefficient. \\
3052 11.2, 11.4--11.11 & Two-sided maximum-modulus bounds prove exact order
3053 and type. The indicator limsup formula holds in every direction, including ties.
3054 Dominant-exponential errors and zero freeness are uniform on compact sets
3055 avoiding ties. Near cancellation and the order-two zero set are checked.
3056 For $N\geq3$, gap and remainder bounds, sector confinement, global exhaustion
3057 and multiplicity identities give the full large-zero-string theorem.
3058 The real leading correction, quadratic exponential error, signed derivative
3059 asymptotic, eventual alternating sign and cubic correction are checked. \\
3060 11.13--11.15 & The reduced series is entire, and
3061 $\Dfun_N(z)=\Delta_N(z^N)$ preserves multiplicity away from zero. Finite
3062 exceptional sets, eventual positive simple reduced zeros, and their stated
3063 displacement are checked, including $N=2$. Both full disk-counting laws
3064 retain all exceptional multiplicities. The full-zero reciprocal sums have
3065 exact thresholds $s>1$ and $s>1/N$, including boundary divergence. \\
3066 12.1 & For every $N\geq2$, the complete multiplicity-indexed product
3067 converges uniformly on compact sets and equals $\Delta_N$. Jensen and Poisson
3068 estimates prove rigidity of the actual quotient from derived growth bounds.
3069 No growth certificate remains. The original-variable product for $\Dfun_N$
3070 is also checked. \\
3071 12.2--12.4 & The actual diagonal operator is trace class, with a
3072 basis-independent trace and determinant given by limits of actual finite
3073 compressions. Infinite complete homogeneous coefficients are limits of finite
3074 monomial sums. Every normalized section has exact order $1/N$. For $N\geq2$
3075 its full zero product and zero-moment Bell formulas are proved. The actual
3076 complex-power germ at zero has the named residue Taylor coefficients. \\
3077 12.6--12.10 & The normalized logarithm on each zero-free disk has the
3078 convergent zero-moment series. Its coefficients equal $\kappa_{N,m}$ and
3079 satisfy Newton's recurrence. The Mittag--Leffler tangent expansion converges
3080 absolutely and locally uniformly away from poles. Tangent coefficients and
3081 the named reciprocal and theta families have their exact zero-moment Bell
3082 formulas. The full tangent series is identified with the analytic function. \\
3083 13.1--13.4 & Analytic division constructs each simple-zero principal part.
3084 Its leading coefficient is uniquely determined. Removing the principal parts
3085 gives a holomorphic remainder and the exact factorial-scaled Cauchy estimate.
3086 Both leading-term errors are bounded, with the first absent for $r=1$.
3087 Rotation gives the orbit-reduced main term. The secant formula holds as a
3088 ratio limit, with an exponentially vanishing relative error. \\
3089 14.2--14.4, 14.6 & The exact all-zero sums retain analytic multiplicity.
3090 Their sharp absolute-convergence domains, boundary divergence and local
3091 uniform convergence are checked. Both Hurwitz comparisons, entire corrections,
3092 global meromorphic continuations, unique simple poles and exact residues are
3093 proved. The order-two/four models have identically zero corrections. Branch
3094 choices on the finite exceptional part change the continuation by an entire
3095 function and preserve its pole and residue. \\
3096 14.8 & The derivative-weighted series uses every actual simple zero and genuine
3097 logarithms with bounded arguments. Absolute and local uniform convergence and
3098 an entire sum are proved for $N\geq3$. For $N=2$, absolute/local uniform
3099 convergence and holomorphy hold on $\Re s>1$. \\
3100 \bottomrule
3101 \end{longtable}
3102 \endgroup
3103 
3104 The declaration-level theorem map is authoritative for the formal
3105 contract. It records full scoped coverage of all 68
3106 theorem-like records: 68 full, zero partial, and zero unmapped. Rows 143--151 close the final fifteen partial records
3107 without altering their statements or assumptions. Imported standard fact
3108 11.1 remains imported. Formal identities, normalized analytic germs, and
3109 fixed logarithm domain identities retain their respective stated meanings.
3110 Finite computations do not change these proof boundaries.
3111 
3112 \section{Conclusion}\label{sec:conclusion}
3113 
3114 Cyclic projection turns the exponential into a finite differential basis, and
3115 the signed cycle produces the distinguished denominator $\Dfun_N$. Its
3116 reciprocal powers yield Bell and Newton recurrences, an additive parameter
3117 law, Appell systems, and theta polynomials. Reduction by $w=z^N$ then connects
3118 the reciprocal coefficients to a genus zero product and reduced-zero power
3119 sums.
3120 
3121 Classical two term localization, specialized to the regular exponent polygon
3122 and combined with reflection symmetry, places all sufficiently large zeros
3123 on symmetry rays. The displacement, derivative and counting estimates yield
3124 the family-specific Hurwitz comparison for the orbit zero zeta.
3125 
3126 \begingroup
3127 \small
3128 \let\prscbibliography\thebibliography
3129 \renewcommand{\thebibliography}[1]{\prscbibliography{#1}\setlength{\itemsep}{0pt}\setlength{\parsep}{0pt}}
3130 \bibliographystyle{alpha}
3131 \bibliography{references}
3132 \endgroup
3133 
3134 \clearpage
3135 \appendix
3136 \section{Visual atlas}\label{app:visual-atlas}
3137 The twenty-six plates retain the worked models of the separate visual guide. The stable labels for Plates IV.1--IV.26 name mathematical mechanisms and point back to the relevant statements. The plates distinguish exact identities, rational certificates, sampled curves, and inherited formal evidence.
3138 \begin{readerroadmap}[title={Three routes through the atlas}]
3139 For algebra, read Plates IV.3--IV.16. For zero geometry and its consequences, read Plates IV.17--IV.24. Plate IV.25 follows one coefficient through the complete calculation. Plate IV.26 explains the evidence attached to the models.
3140 \end{readerroadmap}
3141 \clearpage
3142 \begingroup
3143 \fontsize{10}{12}\selectfont\setlength{\parskip}{4pt}
3144 \setlength{\abovedisplayskip}{6pt}\setlength{\belowdisplayskip}{6pt}
3145 \input{atlas-pages}
3146 \endgroup
3147 
3148 \end{document}

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