Special Functions

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1 \hypertarget{v11-coupled_growth_full-coupled-contraction-functions-and-their-perron-growth-law}{%
2 \chapter[Coupled growth and Perron type]{Coupled contraction functions and their Perron growth
3 law}\label{app:coupled-growth}\label{v11-coupled_growth_full-coupled-contraction-functions-and-their-perron-growth-law}}
4 
5 \hypertarget{v11-coupled_growth_full-definition-and-exact-coefficient-construction}{%
6 \section{Definition and exact coefficient
7 construction}\label{v11-coupled_growth_full-definition-and-exact-coefficient-construction}}
8 
9 Fix real parameters
10 
11 \[
12 0<q,r,a,b<1.
13 \]
14 
15 Consider the normalized pair
16 
17 \[
18 \boxed{\begin{aligned}
19 U'(z)&=1+U(qz)V(rz),\\
20 V'(z)&=1+U(az)V(bz),\\
21 U(0)&=V(0)=0.
22 \end{aligned}}
23 \]
24 
25 \textbf{Theorem F.1.} There is a unique pair of entire solutions. Both
26 functions are odd and have strictly positive coefficients at every odd
27 degree. Write
28 
29 \[
30 U(z)=\sum_{n\ge0}u_nz^{2n+1},\qquad
31 V(z)=\sum_{n\ge0}v_nz^{2n+1}.
32 \]
33 
34 Then \(u_0=v_0=1\), and, for \(n\ge1\),
35 
36 \[
37 \begin{aligned}
38 u_n&=\frac1{2n+1}\sum_{i=0}^{n-1}
39  q^{2i+1}r^{2(n-1-i)+1}u_i v_{n-1-i},\\
40 v_n&=\frac1{2n+1}\sum_{i=0}^{n-1}
41  a^{2i+1}b^{2(n-1-i)+1}u_i v_{n-1-i}.
42 \end{aligned}
43 \]
44 
45 \textbf{Proof.} Comparing successive coefficients in the differential
46 equations determines the coefficient of degree \(k+1\) from coefficients
47 of degrees at most \(k\). Thus the formal pair is unique. The displayed
48 recurrence constructs an odd formal pair satisfying both equations.
49 Formal uniqueness excludes any additional even coefficients. Strict
50 positivity follows by induction.
51 
52 The elementary bounds
53 
54 \[
55 0<u_n\le2^{-n},\qquad 0<v_n\le2^{-n}
56 \]
57 
58 hold by simultaneous induction. If they hold below \(n\), each product
59 in either recurrence is at most \(2^{-(n-1)}\), and there are \(n\)
60 products. The resulting bound is \(n2^{-(n-1)}/(2n+1)<2^{-n}\). Hence
61 the two series converge on \(|z|<\sqrt2\), and define a holomorphic
62 solution there.
63 
64 Put \(Q=\max(q,r,a,b)<1\). If the pair is already defined on \(|z|<R\),
65 then
66 
67 \[
68 \int_0^z[1+U(qw)V(rw)]\,dw,\qquad
69 \int_0^z[1+U(aw)V(bw)]\,dw
70 \]
71 
72 define holomorphic extensions on \(|z|<R/Q\). The integrands are
73 holomorphic on that disk. On the old disk the new functions agree with
74 the old pair because their derivatives and values at zero agree. On the
75 larger disk, every contracted argument still lies in the old disk, so
76 the extensions satisfy the same equations. Iterating exhausts the
77 complex plane. Any other entire solution agrees with the unique germ and
78 therefore everywhere. \(\square\)
79 
80 In particular,
81 
82 \[
83 \begin{aligned}
84 U(z)&=z+\frac{qr}{3}z^3
85  +\frac{qr(qr\,q^2+ab\,r^2)}{15}z^5+O(z^7),\\
86 V(z)&=z+\frac{ab}{3}z^3
87  +\frac{ab(qr\,a^2+ab\,b^2)}{15}z^5+O(z^7).
88 \end{aligned}
89 \]
90 
91 The factorial coefficients \(A_n=(2n+1)!u_n\) and \(B_n=(2n+1)!v_n\)
92 satisfy
93 
94 \[
95 \begin{aligned}
96 A_n&=\sum_{i=0}^{n-1}\binom{2n}{2i+1}
97  q^{2i+1}r^{2(n-1-i)+1}A_iB_{n-1-i},\\
98 B_n&=\sum_{i=0}^{n-1}\binom{2n}{2i+1}
99  a^{2i+1}b^{2(n-1-i)+1}A_iB_{n-1-i}.
100 \end{aligned}
101 \]
102 
103 Thus \(A_n,B_n\) are polynomials with nonnegative integer coefficients
104 in the four scales. These formulas are exact algebraic constructions.
105 Finite checks of them do not replace the analytic proof.
106 
107 \hypertarget{v11-coupled_growth_full-the-characteristic-exponent-and-its-positive-vector}{%
108 \section{The characteristic exponent and its positive
109 vector}\label{v11-coupled_growth_full-the-characteristic-exponent-and-its-positive-vector}}
110 
111 For \(s>0\), let
112 
113 \[
114 P(s)=\begin{pmatrix}q^s&r^s\\a^s&b^s\end{pmatrix}.
115 \]
116 
117 There is a unique \(\rho>0\) with
118 
119 \[
120 \boxed{(1-q^\rho)(1-b^\rho)=(ar)^\rho.}
121 \]
122 
123 Indeed, the left side strictly increases continuously from zero to one,
124 while the right side strictly decreases continuously from one to zero.
125 This equation is precisely \(\det(I-P(\rho))=0\). Define
126 
127 \[
128 \widetilde w=(r^\rho,1-q^\rho)^T,
129 \qquad
130 w=\frac{\widetilde w}{\max(r^\rho,1-q^\rho)}.
131 \]
132 
133 Both coordinates of \(w\) are positive, \(\max(w_1,w_2)=1\), and direct
134 multiplication gives
135 
136 \[
137 P(\rho)w=w.
138 \]
139 
140 The other eigenvalue is \(q^\rho+b^\rho-1\), whose absolute value is
141 strictly less than one. Consequently
142 
143 \[
144 \boxed{\operatorname{spr}P(\rho)=1.}
145 \]
146 
147 Conversely, a positive matrix of this form having spectral radius one
148 has eigenvalue one and satisfies the determinant equation. This proves
149 the proposed spectral characterization without requiring a numerical
150 eigensolver. A positive left eigenvector is
151 
152 \[
153 \ell=(a^\rho,1-q^\rho)^T,\qquad \ell^TP(\rho)=\ell^T.
154 \]
155 
156 The notation \(w\) for the growth vector is deliberately different from
157 \(v_n\), the coefficients of \(V\).
158 
159 \hypertarget{v11-coupled_growth_full-exact-order-and-explicit-positive-type-bounds}{%
160 \section{Exact order and explicit positive type
161 bounds}\label{v11-coupled_growth_full-exact-order-and-explicit-positive-type-bounds}}
162 
163 For \(R>0\), positivity gives
164 
165 \[
166 M_U(R)=\max_{|z|=R}|U(z)|=U(R),\qquad
167 M_V(R)=V(R).
168 \]
169 
170 Set \(F_1=U\), \(F_2=V\), and \(H_i(R)=\log F_i(R)\). Since
171 \(F_i(R)\ge R\), these logarithms are positive for \(R>1\). The type
172 convention used here is
173 
174 \[
175 \underline\tau_i=\liminf_{R\to\infty}\frac{\log M_{F_i}(R)}{R^\rho},
176 \qquad
177 \overline\tau_i=\limsup_{R\to\infty}\frac{\log M_{F_i}(R)}{R^\rho}.
178 \]
179 
180 \textbf{Theorem F.2.} Both functions have order \(\rho\), positive lower
181 type and finite upper type. More precisely, the explicit constants below
182 satisfy
183 
184 \[
185 0<c_-w_i\le\underline\tau_i\le\overline\tau_i\le c_+w_i<\infty,
186 \qquad i=1,2.
187 \]
188 
189 Moreover,
190 
191 \[
192 \lim_{R\to\infty}\frac{\log\log M_{F_i}(R)}{\log R}=\rho.
193 \]
194 
195 This radial exponent limit is stronger than the limsup used to define
196 order. It does not assert that either type has a limit or that growth is
197 uniform in the angular variable.
198 
199 \hypertarget{v11-coupled_growth_full-explicit-lower-comparison}{%
200 \subsection{Explicit lower
201 comparison}\label{v11-coupled_growth_full-explicit-lower-comparison}}
202 
203 Let
204 
205 \[
206 m=\min(q,r,a,b),\qquad Q=\max(q,r,a,b),
207 \]
208 
209 \[
210 C=\frac{Q}{1-Q},\qquad B=1+\frac2m,
211 \qquad c_-=\frac{\log2}{(B+C)^\rho}.
212 \]
213 
214 We prove the stronger statement
215 
216 \[
217 \boxed{H_i(R)\ge c_-w_i(R+C)^\rho\quad(R\ge2,\ i=1,2).}
218 \]
219 
220 On \(2\le R\le B\), its right side is at most \(\log2\), because
221 \(w_i\le1\), whereas \(H_i(R)\ge\log R\ge\log2\).
222 
223 For the first equation, integrate only over the final unit interval to
224 obtain, when \(R\ge B\),
225 
226 \[
227 U(R)\ge\int_{R-1}^R U(qt)V(rt)\,dt
228 \ge U(q(R-1))V(r(R-1)).
229 \]
230 
231 The second equation gives the same inequality with \((q,r)\) replaced by
232 \((a,b)\) and \(U(R)\) replaced by \(V(R)\). All four child radii are at
233 least two. Every scale \(p\in\{q,r,a,b\}\) satisfies
234 
235 \[
236 p(R-1)+C\ge p(R+C),
237 \]
238 
239 because \((1-p)C\ge p\). If the comparison already holds at the child
240 radii, then
241 
242 \[
243 \begin{aligned}
244 H_1(R)&\ge c_-\{w_1[q(R-1)+C]^\rho
245                      +w_2[r(R-1)+C]^\rho\}\\
246 &\ge c_-(q^\rho w_1+r^\rho w_2)(R+C)^\rho
247 =c_-w_1(R+C)^\rho.
248 \end{aligned}
249 \]
250 
251 The second row of \(P(\rho)w=w\) proves the second component in exactly
252 the same way.
253 
254 For a formal propagation, put
255 
256 \[
257 d=(1-Q)B+Q>0.
258 \]
259 
260 Every child of \(R\ge B\) is at most \(Q(R-1)\le R-d\). Simultaneous
261 induction over the intervals \([B+kd,B+(k+1)d]\), starting with the
262 established seed interval, therefore proves both comparisons for every
263 \(R\ge2\). Continuity handles shared endpoints. The argument uses only
264 monotonicity of positive powers and works unchanged when \(0<\rho<1\).
265 
266 \hypertarget{v11-coupled_growth_full-explicit-upper-comparison}{%
267 \subsection{Explicit upper
268 comparison}\label{v11-coupled_growth_full-explicit-upper-comparison}}
269 
270 Set
271 
272 \[
273 \Delta=\log(1/m),\qquad
274 D=\max\left(\log\frac1{qr},\log\frac1{ab}\right)+\log2.
275 \]
276 
277 Choose any integer \(N\ge0\) satisfying \(Q^{-N}\ge3/m\), and define
278 
279 \[
280 W_N=(2^{N+1}-1)\log2
281  +(2^{N+1}-N-2)\log(1/Q),
282 \]
283 
284 \[
285 S=W_N+1+\Delta+D,\qquad
286 c_+=\frac{e^{-\rho}S}{\min(w_1,w_2)}.
287 \]
288 
289 These constants require only the parameters and the uniquely defined
290 root \(\rho\). They require no evaluation of an unknown solution.
291 
292 To bound a seed interval, the coefficient majorant from Theorem F.1 gives
293 \(F_i(1)\le2\). Put \(R_k=Q^{-k}\), \(Z_0=2\), and
294 
295 \[
296 Z_k=Q^{-k}(1+Z_{k-1}^2).
297 \]
298 
299 Monotonicity of the positive functions and the integral equations give
300 \(F_i(R_k)\le Z_k\) simultaneously. As \(Z_k\ge1\),
301 
302 \[
303 \log Z_k\le k\log(1/Q)+\log2+2\log Z_{k-1}.
304 \]
305 
306 Summing this elementary recurrence gives \(\log Z_N\le W_N\). Therefore
307 \(H_i(e^t)\le W_N\) throughout \(1\le t\le1+\Delta\), since
308 \(e^t\le e/m<3/m\le R_N\).
309 
310 Now let
311 
312 \[
313 G_i(t)=H_i(e^t)+t+D.
314 \]
315 
316 For \(t\ge1+\Delta\), all child radii are at least \(e\), so their
317 function values are at least one. The full integral gives
318 
319 \[
320 H_1(e^t)\le t+\log2
321  +H_1(e^{t-\log(1/q)})+H_2(e^{t-\log(1/r)}).
322 \]
323 
324 The definition of \(D\) converts this into
325 
326 \[
327 G_1(t)\le G_1(t-\log(1/q))+G_2(t-\log(1/r)).
328 \]
329 
330 Similarly,
331 
332 \[
333 G_2(t)\le G_1(t-\log(1/a))+G_2(t-\log(1/b)).
334 \]
335 
336 On the seed interval, \(G_i(t)\le S\le c_+w_i e^{\rho t}\). The Perron
337 equation propagates this comparison: substituting the bounds at the
338 children in the first inequality gives
339 \(c_+e^{\rho t}(q^\rho w_1+r^\rho w_2)=c_+w_1e^{\rho t}\), and the
340 second row works likewise. All logarithmic delays belong to
341 \([\log(1/Q),\Delta]\). Induction in intervals of length
342 \(\log(1/Q)>0\), after the seed interval, therefore proves
343 
344 \[
345 \boxed{H_i(R)\le c_+w_iR^\rho-\log R-D\quad(R\ge e).}
346 \]
347 
348 The lower and upper comparisons imply the asserted positive, finite
349 types. Taking logarithms of these bounds for sufficiently large \(R\)
350 gives the claimed radial exponent limits and hence exact orders.
351 \(\square\)
352 
353 \hypertarget{v11-coupled_growth_full-the-two-type-vectors-have-exactly-the-perron-proportions}{%
354 \section{The two type vectors have exactly the Perron
355 proportions}\label{v11-coupled_growth_full-the-two-type-vectors-have-exactly-the-perron-proportions}}
356 
357 The explicit constants above are conservative. There is nevertheless an
358 exact statement about the relative types of the two components.
359 
360 \textbf{Theorem F.3.} There exist positive finite constants
361 \(\underline\tau\le\overline\tau\) such that
362 
363 \[
364 \boxed{\begin{aligned}
365 (\underline\tau_1,\underline\tau_2)^T&=\underline\tau\,w,\\
366 (\overline\tau_1,\overline\tau_2)^T&=\overline\tau\,w.
367 \end{aligned}}
368 \]
369 
370 In particular,
371 
372 \[
373 \frac{\underline\tau_1}{\underline\tau_2}
374 =\frac{\overline\tau_1}{\overline\tau_2}
375 =\frac{r^\rho}{1-q^\rho}
376 =\frac{1-b^\rho}{a^\rho}.
377 \]
378 
379 Either both ordinary type limits exist or neither does. When they exist,
380 their vector is a positive multiple of \(w\).
381 
382 \textbf{Proof.} Use the final-unit integral inequality from the lower
383 proof and divide by \(R^\rho\). Taking the liminf gives
384 
385 \[
386 \underline t\ge P(\rho)\underline t,
387 \qquad
388 \underline t=(\underline\tau_1,\underline\tau_2)^T.
389 \]
390 
391 Here \(p(R-1)\) tends to infinity continuously and
392 \([p(R-1)]^\rho/R^\rho\to p^\rho\), which justifies each scaled liminf.
393 The liminf of a sum is at least the sum of the liminfs. All quantities
394 are positive and finite by Theorem F.2.
395 
396 The full-integral inequality
397 
398 \[
399 H_1(R)\le\log R+\log2+H_1(qR)+H_2(rR)
400 \]
401 
402 and its second-row counterpart imply, on taking the limsup,
403 
404 \[
405 \overline t\le P(\rho)\overline t.
406 \]
407 
408 The terms \((\log R+\log2)/R^\rho\) vanish. Dotting either componentwise
409 inequality with the strictly positive left eigenvector \(\ell\) from
410 Section~\ref{v11-coupled_growth_full-the-characteristic-exponent-and-its-positive-vector} shows that its nonnegative difference vector has zero positive
411 weighted sum. Every coordinate of that difference is therefore zero.
412 Thus both type vectors are fixed by \(P(\rho)\). Its eigenvalue-one
413 space is one dimensional, so each is a positive multiple of \(w\).
414 
415 If one component has an ordinary type limit, its lower and upper types
416 are equal. Because that coordinate of \(w\) is positive, this forces
417 \(\underline\tau=\overline\tau\), and the other component also has a
418 type limit. \(\square\)
419 
420 This theorem does not assert a limit for \(\log U(R)/\log V(R)\) when
421 type oscillations persist. It also does not identify the scalar values
422 of either type or prove their equality.
423 
424 \hypertarget{v11-coupled_growth_full-reductions-a-worked-unequal-pair-and-parameter-structure}{%
425 \section{Reductions, a worked unequal pair, and parameter
426 structure}\label{v11-coupled_growth_full-reductions-a-worked-unequal-pair-and-parameter-structure}}
427 
428 \textbf{Exact scalar reduction.} The components coincide if and only if
429 the unordered pairs of scales coincide:
430 
431 \[
432 U=V\quad\Longleftrightarrow\quad\{q,r\}=\{a,b\}.
433 \]
434 
435 For sufficiency, the scalar solution \(T_{q,r}\) from
436 \path{routes/asymmetric_dilation.md} makes \((T_{q,r},T_{q,r})\)
437 solve the coupled equations. Theorem F.1 gives uniqueness. For necessity,
438 equality at degree three gives \(qr=ab=p>0\). Equality at degree five
439 then gives \(q^2+r^2=a^2+b^2\). The two positive unordered pairs have
440 the same product and the same sum \(\sqrt{q^2+r^2+2p}\), so they
441 coincide. This characterizes equality of the normalized components. A
442 stronger assertion about all possible changes of variables requires its
443 own nonreduction theorem.
444 
445 On either scalar reduction locus, the characteristic equation is
446 equivalent to \(q^\rho+r^\rho=1\), recovering the scalar order theorem.
447 At \(q=r=a=b=1/2\),
448 
449 \[
450 U(z)=V(z)=\sqrt2\sinh(z/\sqrt2),\qquad \rho=1.
451 \]
452 
453 \textbf{A genuinely unequal order-one pair.} Set
454 
455 \[
456 (q,r,a,b)=\left(\frac15,\frac25,\frac35,\frac7{10}\right).
457 \]
458 
459 The determinant identity at \(\rho=1\) is
460 \((1-1/5)(1-7/10)=(3/5)(2/5)=6/25\). Thus the exact order is one, and
461 
462 \[
463 w=(1/2,1)^T.
464 \]
465 
466 The first terms are
467 
468 \[
469 \begin{aligned}
470 U(z)&=z+\frac2{75}z^3+\frac{88}{234375}z^5
471  +\frac{18814}{8544921875}z^7+O(z^9),\\
472 V(z)&=z+\frac7{50}z^3+\frac{8211}{1250000}z^5
473  +\frac{85664601}{625000000000}z^7+O(z^9).
474 \end{aligned}
475 \]
476 
477 Both lower and upper exponential types of \(V\) are exactly twice the
478 corresponding types of \(U\). The row sums of \(P(1)\) are \(3/5\) and
479 \(13/10\). Consequently neither an unweighted average row condition nor
480 the maximum of the two scalar row orders yields this example's coupled
481 order. The positive eigenvector accounts for the exchange of growth
482 between the components.
483 
484 For this example the elementary lower constants are \(m=1/5\),
485 \(Q=7/10\), \(C=7/3\), \(B=11\), and \(c_-=3\log2/40\). Hence, for every
486 \(R\ge2\),
487 
488 \[
489 \log U(R)\ge\frac{3\log2}{80}\left(R+\frac73\right),\qquad
490 \log V(R)\ge\frac{3\log2}{40}\left(R+\frac73\right).
491 \]
492 
493 \textbf{Dependence of the order on the scales.} The determinant equation
494 shows that the order depends on the self scales \(q,b\) and the product
495 \(ar\) of the two cross scales. Changing \(a\) and \(r\) while keeping
496 their product fixed leaves the order fixed, provided both remain in
497 \((0,1)\). The Perron vector and the normalized functions generally
498 change.
499 
500 For any prescribed \(s>0\) and any \(q,r\in(0,1)\), choices satisfying
501 
502 \[
503 a^s r^s=(1-q^s)(1-b^s),\qquad 0<a,b<1,
504 \]
505 
506 lie on the order-\(s\) surface. For instance, one may select \(q,r,b\)
507 with \((1-q^s)(1-b^s)<r^s\), then define
508 
509 \[
510 a=\frac{[(1-q^s)(1-b^s)]^{1/s}}{r}.
511 \]
512 
513 This constructs exact order surfaces without fitting growth from
514 truncated series.
515 

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