Special Functions

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1 \hypertarget{exact-sequence-polyphase-and-tap-laboratory}{%
2 \chapter[Sequence and polyphase laboratory]{Exact sequence, polyphase and tap
3 laboratory}\label{app:sequence}\label{exact-sequence-polyphase-and-tap-laboratory}}
4 
5 
6 
7 This chapter connects the nonlinear family to exact sequences and finite
8 filter algebra. Its reusable code implements finite calculations. The
9 analytic results are proved in Chapter~\ref{ch:feedback} and
10 Appendix~\ref{app:growth}. All mathematical
11 sequence indices begin at zero. All finite arrays are zero extended
12 beyond their displayed support. The catalog contains explicit candidates
13 for OEIS lookup. This chapter does not assert an OEIS accession or
14 historical priority for an unsearched row.
15 
16 \hypertarget{the-integer-polynomial-object}{%
17 \section{The integer polynomial
18 object}\label{the-integer-polynomial-object}}
19 
20 Write the unique odd formal solution as
21 
22 \[
23 T_{q,r}(z)=\sum_{n\ge0}c_n(q,r)z^{2n+1}
24           =\sum_{n\ge0}A_n(q,r)\frac{z^{2n+1}}{(2n+1)!}.
25 \]
26 
27 The two coefficient systems are related by \(A_n=(2n+1)!c_n\). In
28 particular, \(A_n\) is the derivative of order \(2n+1\) at zero. The
29 coefficient \(c_n\), the derivative \(A_n\), and the full EGF sequence
30 \(a_m\) with \(a_{2n+1}=A_n, a_{2n}=0\) are different objects. None of
31 the code silently substitutes one for another.
32 
33 The nonlinear differential equation gives the recurrence
34 
35 \[
36 \boxed{
37 A_0=1,\qquad
38 A_n=\sum_{i=0}^{n-1}\binom{2n}{2i+1}
39  q^{2i+1}r^{2(n-1-i)+1}A_i A_{n-1-i},\quad n\ge1.}
40 \]
41 
42 \textbf{Proof.} Multiply the ordinary Taylor recurrence by \((2n+1)!\).
43 The factor attached to \(A_iA_j\), where \(i+j=n-1\), is
44 \((2n)!/[(2i+1)!(2j+1)!]=\binom{2n}{2i+1}\). This also proves the
45 recurrence directly in the EGF convention. \(\square\)
46 
47 \textbf{Polynomial theorem.} For every \(n\), \(A_n\in\mathbb Z[q,r]\),
48 all its nonzero coefficients are positive integers, and
49 \(A_n(q,r)=A_n(r,q)\). Every monomial \(q^u r^v\) satisfies
50 
51 \[
52 u,v\ge n,\qquad u\equiv v\equiv n\pmod2,\qquad u+v\le n(n+1).
53 \]
54 
55 \textbf{Proof.} The recurrence is an induction using positive integer
56 binomial coefficients. Swapping \(q,r\) and replacing \(i\) by \(n-1-i\)
57 leaves the sum unchanged. If child indices are \(i,j\), their exponent
58 pairs receive the shifts \((2i+1,2j+1)\). The lower bounds follow from
59 \(i+j+1=n\). The two parities are the parity of \(i+j+1\), also \(n\).
60 For the degree bound, a recurrence term has total degree at most
61 
62 \[
63 2n+i(i+1)+j(j+1)\le n(n+1),\qquad i+j=n-1,
64 \]
65 
66 since \(i(i+1)+j(j+1)\) is largest at \((i,j)=(0,n-1)\) or its swap. The
67 base \(A_0=1\) satisfies every assertion. \(\square\)
68 
69 The first four polynomials are
70 
71 \[
72 \begin{aligned}
73 A_0={}&1,\\
74 A_1={}&2qr,\\
75 A_2={}&8q^2r^2(q^2+r^2),\\
76 A_3={}&q^3r^3\bigl[48(q^2+r^2)(q^4+r^4)+80q^2r^2\bigr].
77 \end{aligned}
78 \]
79 
80 Consequently each integer pair \((q,r)\) gives an integer sequence for
81 all \(n\), independently of whether its formal Taylor series converges.
82 For example, the first values of \(A_n(1,2)\) are
83 \(1,4,160,35200,38041600\).
84 
85 \hypertarget{rational-feedback-banks-also-admit-integer-sequences}{%
86 \subsection{Rational feedback banks also admit integer
87 sequences}\label{rational-feedback-banks-also-admit-integer-sequences}}
88 
89 For rational channel tuples \((h_t,q_t,r_t)\), use
90 
91 \[
92 F'(z)=1+\sum_t h_tF(q_tz)F(r_tz),\qquad F(0)=0.
93 \]
94 
95 The preceding recurrence is summed over channels with the factor
96 \(h_t\). In the universal polynomial over the weights and contractions,
97 every term of \(A_n\) has weight degree exactly \(n\) and total
98 contraction degree at most \(n(n+1)\). The weight statement follows from
99 one new weight at each recurrence step and \(1+i+j=n\); the contraction
100 bound uses the same induction.
101 
102 Let \(H\) be a common denominator for all channel weights, and \(D\) a
103 common denominator for all channel contractions. Then
104 
105 \[
106 \boxed{B_n=H^nD^{n(n+1)}A_n\in\mathbb Z\quad\hbox{for all }n.}
107 \]
108 
109 This is a proved normalization, even with signed rational weights. It is
110 usually not the smallest possible denominator clearing. The catalog
111 records both \(A_n\) and \(B_n\), and the exact multiplier, so that a
112 proposed sequence lookup can state its normalization honestly.
113 
114 \hypertarget{a-sharp-distinction-between-sequences-and-analytic-germs}{%
115 \section{A sharp distinction between sequences and analytic
116 germs}\label{a-sharp-distinction-between-sequences-and-analytic-germs}}
117 
118 The entire-function theorem uses \(0<q,r<1\). Specializations with large
119 positive integers produce valid formal and integer sequences but do not
120 necessarily produce analytic functions near zero.
121 
122 \textbf{Zero-radius theorem.} If \(q>0\) and \(r>1\), the formal series
123 for \(T_{q,r}\) has radius of convergence zero. The same holds after
124 swapping the parameters.
125 
126 \textbf{Proof.} Every coefficient is positive. In the ordinary
127 recurrence keep only the term \(i=0\). This gives
128 
129 \[
130 c_n\ge\frac{q r^{2n-1}}{2n+1}c_{n-1}
131 \quad\Longrightarrow\quad
132 c_n\ge\frac{q^n r^{n^2}}{3\cdot5\cdots(2n+1)}.
133 \]
134 
135 Since the denominator is at most \((2n+1)^n\),
136 
137 \[
138 c_n^{1/(2n+1)}\ge
139 q^{n/(2n+1)}r^{n^2/(2n+1)}(2n+1)^{-n/(2n+1)}\longrightarrow\infty.
140 \]
141 
142 The Cauchy--Hadamard criterion therefore gives radius zero. \(\square\)
143 
144 More generally, a positive feedback bank with a channel satisfying
145 \(h>0,q>0,r>1\) has the same obstruction, replacing \(q^n\) by
146 \((hq)^n\). Positivity prevents cancellation. No conclusion of this form
147 is asserted for a bank with signed weights.
148 
149 \begin{longtable}[]{@{}
150   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
151   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}
152   >{\raggedright\arraybackslash}P{(\columnwidth - 4\tabcolsep) * \real{0.3333}}@{}}
153 \toprule\noalign{}
154 \begin{minipage}[b]{\linewidth}\raggedright
155 Parameters or bank
156 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
157 First five \(A_n\)
158 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
159 Mathematical status
160 \end{minipage} \\
161 \midrule\noalign{}
162 \endhead
163 \bottomrule\noalign{}
164 \endlastfoot
165 \((1,1)\) & \(1\), \(2\), \(16\), \(272\), \(7936\) & Known tangent derivatives \\
166 \((0,1)\) & \(1\), \(0\), \(0\), \(0\), \(0\) & Known linear boundary \\
167 \((1,-1)\) & \(1\), \(-2\), \(16\), \(-272\), \(7936\) & Known hyperbolic tangent
168 derivatives \\
169 \((1,2)\) & \(1\), \(4\), \(160\), \(35200\), \(38041600\) & Integer sequence, radius
170 zero \\
171 \((1/2,1/2)\) & \(1\), \(1/2\), \(1/4\), \(1/8\), \(1/16\) & Known scaled hyperbolic sine \\
172 \((1/4,3/4)\) & \(1\), \(3/8\), \(45/256\), \(21465/262144\), \(39833775/1073741824\) &
173 Entire asymmetric family, order one \\
174 \end{longtable}
175 
176 The complete catalog contains \textbf{18 rows}, each through \(n=12\):
177 three classical or degenerate integer anchors, eight positive integer
178 formal specializations, four rational contractions, one entire edge case
179 of infinite order, and two normalized feedback banks. Integer
180 integrality and rational clearing are established by the preceding
181 proofs, not merely inferred from the finite sample.
182 
183 \hypertarget{the-uncontracted-boundary-is-entire-of-infinite-order}{%
184 \subsection{The uncontracted boundary is entire of infinite
185 order}\label{the-uncontracted-boundary-is-entire-of-infinite-order}}
186 
187 \textbf{Boundary theorem.} For \(0<q<1\), the normalized formal solution
188 of
189 
190 \[
191 T'(z)=1+T(qz)T(z),\qquad T(0)=0
192 \]
193 
194 extends to an entire function of infinite order. The swapped boundary
195 has the same property.
196 
197 \textbf{Proof of entire extension.} The tangent-coefficient majorant
198 gives an initial holomorphic germ. Suppose \(T\) is known on \(|z|<R\).
199 The function \(T(qz)\) is known on \(|z|<R/q\). On this larger disk
200 solve the linear ODE \(U'=1+T(qz)U\), \(U(0)=0\). Explicitly,
201 
202 \[
203 U(z)=e^{B(z)}\int_0^z e^{-B(w)}\,dw,
204 \qquad B(z)=\int_0^zT(qw)\,dw.
205 \]
206 
207 The integrals are path independent on the disk. Linear-ODE uniqueness
208 shows \(U=T\) on the previous disk, and there \(T(qz)\) equals the newly
209 extended function at \(qz\). Thus the same nonlinear equation holds on
210 the enlarged disk. Iterating \(R\mapsto R/q\) exhausts the plane.
211 
212 \textbf{Proof of infinite order.} Positive coefficients imply
213 \(T(x)\ge x\) for \(x>0\) and \(M(x)=T(x)\). The logarithmic derivative
214 obeys
215 
216 \[
217 \frac{d}{dx}\log T(x)=\frac1{T(x)}+T(qx)\ge qx.
218 \]
219 
220 Integrating from one, with \(T(1)\ge1\), gives
221 \(T(x)\ge\exp(q(x^2-1)/2)\) for \(x\ge1\). For \(R\ge2/q\), integrate
222 the same logarithmic derivative over \([R/2,R]\). Since \(T(R/2)\ge1\),
223 
224 \[
225 \boxed{
226 \log M(R)\ge\frac R2\exp\left(\frac{q^3R^2}{8}-\frac q2\right).}
227 \]
228 
229 Therefore \(\log\log M(R)/\log R\to\infty\), which proves infinite
230 order. \(\square\)
231 
232 For nonnegative parameters this gives a complete analytic/formal phase
233 chart:
234 
235 \begin{longtable}[]{@{}
236   >{\raggedright\arraybackslash}P{(\columnwidth - 2\tabcolsep) * \real{0.5000}}
237   >{\raggedright\arraybackslash}P{(\columnwidth - 2\tabcolsep) * \real{0.5000}}@{}}
238 \toprule\noalign{}
239 \begin{minipage}[b]{\linewidth}\raggedright
240 Parameter region
241 \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
242 Normalized solution
243 \end{minipage} \\
244 \midrule\noalign{}
245 \endhead
246 \bottomrule\noalign{}
247 \endlastfoot
248 \(0<q,r<1\) & Entire, finite order defined by \(q^\rho+r^\rho=1\) \\
249 One parameter equals one, the other lies in \((0,1)\) & Entire, infinite
250 order \\
251 \(q=r=1\) & Tangent, meromorphic with its usual poles \\
252 \(q>0,r>0,\max(q,r)>1\) & Formal solution with radius zero \\
253 One parameter equals zero & Linear solution \(z\), regardless of the
254 other scale \\
255 \end{longtable}
256 
257 The finite-order formula applies in the open contraction square. It does
258 not assign a finite order to the edge, where its defining scale equation
259 has no positive root.
260 
261 \hypertarget{exact-root-filter-sections}{%
262 \section{Exact root-filter
263 sections}\label{exact-root-filter-sections}}
264 
265 For an EGF \(G(z)=\sum_{m\ge0}a_m z^m/m!\), an integer base \(b\ge1\), a
266 residue \(0\le j<b\), and \(\omega=e^{2\pi i/b}\), define
267 
268 \[
269 \boxed{
270 \mathcal P_{b,j}G(z)=\frac1b\sum_{k=0}^{b-1}\omega^{-jk}G(\omega^kz)
271 =\sum_{m\equiv j\,(b)}a_m\frac{z^m}{m!}.}
272 \]
273 
274 This holds as a formal series and wherever the original series
275 converges. Indeed the coefficient multiplier is the geometric sum
276 \(b^{-1}\sum_{k=0}^{b-1}\omega^{k(m-j)}\), equal to one or zero
277 according to the residue. It proves all the projector identities
278 
279 \[
280 \sum_{j=0}^{b-1}\mathcal P_{b,j}=I,\qquad
281 \mathcal P_{b,j}\mathcal P_{b,l}=\delta_{jl}\mathcal P_{b,j}.
282 \]
283 
284 The full-position section and the decimated lane are distinct arrays. A
285 section keeps the original positions, inserting zeros outside the
286 residue. A lane \(x_j[n]=a_{bn+j}\) changes the index to \(n\).
287 
288 For an \textbf{ordinary} generating function \(X(z)=\sum_ma_m z^m\),
289 define \(X_j(w)=\sum_na_{bn+j}w^n\). Then
290 
291 \[
292 X(z)=\sum_{j=0}^{b-1}z^jX_j(z^b).
293 \]
294 
295 For an EGF the corresponding exact reconstruction is
296 
297 \[
298 G(z)=\sum_{j=0}^{b-1}z^j
299  \sum_{n\ge0}\frac{a_{bn+j}}{(bn+j)!}(z^b)^n.
300 \]
301 
302 One must retain \((bn+j)!\). Replacing it by \(n!\) gives a different
303 function. The root-filter section identity has no such factorial
304 ambiguity.
305 
306 \hypertarget{ordinary-taps-egf-multiplication-and-differentiation}{%
307 \section{Ordinary taps, EGF multiplication and
308 differentiation}\label{ordinary-taps-egf-multiplication-and-differentiation}}
309 
310 Let \(h_0,\ldots,h_{L-1}\) be a finite tap vector and let \(a_m=0\) for
311 \(m<0\). Ordinary taps form
312 
313 \[
314 b_m=\sum_{t=0}^{L-1}h_ta_{m-t}.
315 \]
316 
317 For OGFs this is multiplication by \(H(z)=\sum_th_tz^t\):
318 \(B(z)=H(z)X(z)\). For EGFs it is a sum of repeated zero-constant
319 antiderivatives:
320 
321 \[
322 \boxed{B_{\rm EGF}(z)=\sum_{t=0}^{L-1}h_t I^tG(z),\qquad
323 IG(z)=\int_0^zG(w)\,dw.}
324 \]
325 
326 \textbf{Proof.} The EGF of the delayed sequence is
327 \(\sum_{m\ge t}a_{m-t}z^m/m!=I^tG(z)\). Summing taps gives the formula.
328 Conversely, \(G^{(t)}\) has coefficient sequence \(a_{m+t}\), so
329 derivatives sample future coefficients and do not implement these past
330 delays. \(\square\)
331 
332 For two EGFs \(G_a,G_h\), ordinary function multiplication instead gives
333 the \textbf{Hurwitz product}
334 
335 \[
336 G_a(z)G_h(z)=\sum_m\left[
337 \sum_{t=0}^m\binom mt h_ta_{m-t}\right]\frac{z^m}{m!}.
338 \]
339 
340 This is a binomial convolution. For example, with \(a_m=1\) and taps
341 \(h=(1,1)\), ordinary taps give \(b_2=2\), while the Hurwitz product
342 gives \(b_2=3\). This explicit negative control catches a common
343 normalization error.
344 
345 \textbf{Intrinsic nonlinear feedback taps} are the parameters
346 \((h_t,q_t,r_t)\) inside \(F'=1+\sum_th_tF(q_tz)F(r_tz)\). They change
347 the function's defining equation and its derivative recurrence. Ordinary
348 FIR taps are a linear transformation of an already defined coefficient
349 sequence. Both are useful, but they operate on different objects. The
350 intrinsic-bank growth theorem is proved in Appendix~\ref{app:growth}.
351 
352 \hypertarget{the-exact-finite-polyphase-tap-matrix}{%
353 \section{The exact finite polyphase tap
354 matrix}\label{the-exact-finite-polyphase-tap-matrix}}
355 
356 For an OGF sequence \(a_m\), define lanes \(x_l[n]=a_{bn+l}\) and
357 \(X_l(w)=\sum_nx_l[n]w^n\). Apply finite ordinary taps and obtain
358 \(y_j[n]=b_{bn+j}\). Then the exact matrix identity is
359 
360 \[
361 \boxed{Y_j(w)=\sum_{l=0}^{b-1}H_{jl}(w)X_l(w),\qquad
362 H_{jl}(w)=\sum_{\substack{0\le t<L\\t+l\equiv j\,(b)}}
363 h_t w^{(t+l-j)/b}.}
364 \]
365 
366 Every exponent is a nonnegative integer. It records the carry across
367 phase boundaries, rather than erasing it.
368 
369 \textbf{Proof.} A term \(h_t a_{bn+j-t}\) is in phase \(l\) exactly when
370 \(t+l\equiv j\pmod b\). Write \(d=(t+l-j)/b\). That term is
371 \(h_t x_l[n-d]\), whose OGF is \(h_tw^dX_l(w)\) by zero extension.
372 Summing over taps and phases gives the matrix. \(\square\)
373 
374 For two phases and two taps,
375 
376 \[
377 \begin{pmatrix}Y_0\\Y_1\end{pmatrix}
378 =\begin{pmatrix}h_0&h_1w\\h_1&h_0\end{pmatrix}
379 \begin{pmatrix}X_0\\X_1\end{pmatrix}.
380 \]
381 
382 The upper-right factor \(w\) is necessary: a delay from odd phase to
383 even phase crosses one block. A diagonal-only phase operation cannot
384 implement this filter. The code constructs the sparse matrix for any
385 finite base and tap vector and checks that recombination gives the full
386 direct convolution.
387 
388 \hypertarget{reuse-and-evidence}{%
389 \section{Reuse and evidence}\label{reuse-and-evidence}}
390 
391 From the package\textquotesingle s \texttt{research/} directory, standard Python is enough:
392 
393 \begin{verbatim}
394 python family_lab.py coefficients --q 1/4 --r 3/4 --n 8
395 python family_lab.py catalog --n 12 --output-dir examples
396 python checks/family_lab_checks.py
397 \end{verbatim}
398 
399 The reusable functions include \path{asymmetric_odd_derivatives},
400 \path{feedback_odd_derivatives}, \path{asymmetric_polynomials},
401 \texttt{section}, \path{phase_lanes}, \path{merge_lanes},
402 \path{ordinary_taps}, \path{hurwitz_product},
403 \path{derivative_shift}, \path{integral_shift},
404 \path{polyphase_tap_matrix}, and \path{apply_polyphase_matrix}.
405 All core arithmetic is \texttt{fractions.Fraction} or integer
406 arithmetic. The phase and tap API applies to any finite rational
407 sequence, not just the special-function catalog.
408 
409 The final completed run passed \textbf{8,038 exact assertions} and is
410 recorded in \path{evidence/family_lab_checks.json}. Its scope
411 includes:
412 
413 \begin{itemize}
414 \tightlist
415 \item
416   18 catalog rows through \(A_{12}\), independently compared with dense
417   formal Picard iteration through degree 25.
418 \item
419   Bivariate polynomials through \(A_8\), with 540 nonzero monomials
420   checked for integer positivity, symmetry, parity and the degree bound.
421 \item
422   Tangent coefficients independently obtained by division of sine and
423   cosine, plus hyperbolic tangent, scaled hyperbolic sine and linear anchors.
424 \item
425   Root-filter multipliers reduced exactly in cyclotomic fields for bases
426   one through seven and indices zero through 31.
427 \item
428   Direct FIR convolution compared with phase-matrix convolution for 26
429   input lengths, seven bases and four tap vectors, including signed
430   rational taps.
431 \item
432   The antiderivative interpretation of EGF taps and the factorial
433   conversion of Hurwitz products, independently compared with ordinary
434   Taylor products.
435 \item
436   Negative controls for omitted phase carries, aliasing lanes by
437   concatenation, silent truncation, derivative/delay confusion and
438   Hurwitz/FIR confusion.
439 \end{itemize}
440 
441 The bounded checks validate the implemented paths in the recorded
442 ranges. The universal polynomial, degree, clearing, root-filter and
443 phase-matrix statements follow from the proofs above. Entire continuation and growth order require the analytic proofs, not these
444 finite checks. Historical identification of a catalog row requires a
445 separate literature comparison.
446 
447 \path{evidence/family_lab_checks.json} records software version,
448 code hashes, exact arithmetic, bounds, category counts and run time. It
449 reports no failed assertion in this run. Independent audit identified
450 two API input-validation defects: a fractional residue was accepted by
451 the section operator, and an empty phase matrix accepted base zero. Both
452 are now rejected and have passing regression checks. These defects
453 affected invalid inputs, not the displayed identities. A draft prose
454 example initially transcribed \(A_3(1,2)\) and \(A_4(1,2)\) incorrectly;
455 the exact catalog comparison exposed the mismatch and corrected them to
456 \(35200\) and \(38041600\). The code result did not change. Larger
457 numerical evidence is unnecessary for these algebraic identities; an
458 independent mathematical audit should examine the reductions.

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