Special Functions

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1 \pageheading{EXACT LABORATORY\; /\; COEFFICIENTS}{Integer coordinates for the nonlinear family}
2 Normalize the Taylor coefficients by
3 \[
4 A_n(q,r)=(2n+1)!c_n(q,r),\qquad A_0=1.
5 \]
6 The recurrence becomes
7 \[
8 \boxed{A_n=\sum_{i=0}^{n-1}\binom{2n}{2i+1}
9 q^{2i+1}r^{2(n-1-i)+1}A_iA_{n-1-i},\quad n\ge1.}
10 \]
11 Therefore $A_n\in\mathbb Z[q,r]$ has nonnegative coefficients and is
12 symmetric in its two variables. The first coordinates are
13 \[
14 \begin{aligned}
15 A_1&=2qr,\\
16 A_2&=8q^2r^2(q^2+r^2),\\
17 A_3&=q^3r^3[48(q^2+r^2)(q^4+r^4)+80q^2r^2].
18 \end{aligned}
19 \]
20 At $(q,r)=(1,1)$ these are the tangent numbers
21 $1,2,16,272,7936,\ldots$, the known OEIS A000182 anchor.
22 At $(1,2)$ the exact sequence begins
23 $1,4,160,35200,38041600,\ldots$.
24 \smallfig{17_tree_addresses}{Ordered trees retain left/right path information.
25 The two displayed shapes contribute different monomials to the same
26 factorial-normalized coefficient. Their labels are exact tree data.}
27 At rational scales, an integer denominator clearing is available. If $D$
28 clears every contraction denominator and $H$ clears every bank weight,
29 \[
30 H^nD^{n(n+1)}A_n\in\mathbb Z.
31 \]
32 The maximum root-path exponent sum over a full binary tree is $n(n+1)$.
33 The clearing provides exact sequence exports without implying a new OEIS entry.
34 
35 \pageheading{EXACT LABORATORY\; /\; REGIMES}{A formal sequence is not always an analytic germ}
36 \smallfig[0.68]{18_parameter_regimes}{The full nonnegative real parameter map separates
37 finite-order entire functions, infinite-order entire edges, tangent, and
38 zero-radius formal series. Zero scales give the linear function.}
39 If $q>0$, $r>1$, the recurrence includes its $i=0$ term and gives
40 \[
41 c_n\ge\frac{q^nr^{n^2}}{3\cdot5\cdots(2n+1)}.
42 \]
43 The root test proves zero radius. By symmetry this covers either scale above
44 one, provided the other is positive. The exported positive integer asymmetric
45 rows belong to this formal regime.
46 
47 If $0<q<1$, $r=1$, the equation is linear in the uncontracted value:
48 $T'=1+T(qz)T(z)$. A known coefficient on $|z|<R/q$ gives a linear ODE
49 there, so disk bootstrapping proves entire extension. On the positive ray,
50 $(\log T)'=T(qR)+1/T(R)\ge qR$, giving an exponential lower bound for
51 $T$. Substitution back gives, for $R\ge2/q$,
52 \[
53 \log T(R)\ge\frac R2\exp(q^3R^2/8-q/2).
54 \]
55 Thus the entire order is infinite. The corner $(1,1)$ is tangent with poles.
56 These exact boundaries protect the analytic interpretation of the catalog.
57 
58 \pageheading{EXACT LABORATORY\; /\; PROJECTORS}{Polyphase sections preserve the full sequence}
59 For an ordinary generating function $G(z)=\sum_{m\ge0}a_mz^m$ and
60 integer $b\ge1$, write
61 \[
62 G(z)=\sum_{j=0}^{b-1}z^jG_j(z^b),\qquad
63 G_j(t)=\sum_{n\ge0}a_{bn+j}t^n.
64 \]
65 The root projector is
66 \[
67 P_{b,j}G(z)=\frac1b\sum_{k=0}^{b-1}\omega^{-jk}G(\omega^kz)
68 =z^jG_j(z^b),\qquad\omega=e^{2\pi i/b}.
69 \]
70 The same projector works on an EGF, but the section retains its original
71 factorials: $\sum_na_{bn+j}z^{bn+j}/(bn+j)!$.
72 Removing those denominators changes the representation and requires an
73 explicit transport. Root filtering does not do it automatically.
74 
75 For causal FIR taps $h_\ell$, $b_m=\sum_\ell h_\ell a_{m-\ell}$ with
76 zero extension to negative indices, split $A(z)$ and $B(z)$ into $b$ phases.
77 Then $B_i(t)=\sum_jH_{ij}(t)A_j(t)$, where
78 \[
79 \boxed{H_{ij}(t)=\sum_{\substack{\ell\ge0\\j+\ell\equiv i\pmod b}}
80 h_\ell t^{(j+\ell-i)/b}.}
81 \]
82 The exponent is the nonnegative carry to the next phase block. This is an
83 exact coefficient law, with no approximation or loss of terms.
84 
85 \pageheading{EXACT LABORATORY\; /\; TAPS}{The carry matrix is part of the filter}
86 \smallfig{16_polyphase}{A three-phase filter with taps $[1,2,-1]$.
87 The matrix carries terms across block boundaries. Interleaving its output
88 reproduces the direct scalar FIR calculation exactly.}
89 With these taps and $b=3$,
90 \[
91 H(t)=\begin{pmatrix}
92 1&-t&2t\\2&1&-t\\-1&2&1
93 \end{pmatrix}.
94 \]
95 The entries arise from the same congruence condition. For instance an input
96 in phase two shifted by one lands in phase zero of the next block, contributing
97 $2t$ to $H_{0,2}$.
98 
99 \claim{Three operations that must be kept distinct}
100 For an EGF $E_a(z)=\sum_ma_mz^m/m!$, differentiation transports future
101 coefficients: $E_a^{(k)}=\sum_ma_{m+k}z^m/m!$.
102 Causal ordinary taps instead act by repeated zero-constant integration:
103 \[
104 E_b=\sum_\ell h_\ell I^\ell E_a.
105 \]
106 Multiplication of two EGFs gives Hurwitz convolution,
107 $b_m=\sum_{k=0}^m\binom mk a_kh_{m-k}$.
108 It is not ordinary sequence FIR convolution. Intrinsic nonlinear feedback
109 taps alter the defining family and are a fourth operation with their own
110 coefficient recurrence.
111 
112 \pageheading{EXACT LABORATORY\; /\; REUSABLE CODE}{A catalog with explicit provenance}
113 The standard-library \texttt{family\_lab.py} exposes exact coefficient
114 polynomials, normalized sequences, positive-bank coefficients, rational
115 denominator clearing, root sections, Hurwitz products, ordinary FIR taps,
116 and polyphase filter matrices. The catalog contains 18 rows through index 12.
117 
118 Every row records parameters, coefficient normalization, arithmetic regime,
119 and whether its entries are integers or rationals. Eight positive integer
120 asymmetric specializations are labeled zero-radius formal sequences. Entire
121 interior examples and an infinite-order edge are labeled separately.
122 
123 \begin{tabular}{@{}P{0.27\linewidth}P{0.35\linewidth}P{0.30\linewidth}@{}}
124 \toprule
125 Example & First normalized coordinates & Interpretation\\\midrule
126 $(1,1)$ & $1,2,16,272,7936$ & Known tangent anchor\\
127 $(1,2)$ & $1,4,160,35200,38041600$ & Integer, formal radius zero\\
128 $(1/2,1/2)$ & $1,1/2,1/4,1/8,1/16$ & Scaled sinh, entire order one\\
129 $(1/3,2/3)$ & $1,4/9,160/729,18560/177147,\ldots$ & Asymmetric entire order one\\
130 \bottomrule
131 \end{tabular}
132 
133 \claim{Implementation boundaries are enforced}
134 Phase residues and bases must be integers, and a phase bank cannot be empty.
135 The independent review found that fractional residues and zero phases were
136 previously accepted. Both domain-validation defects were fixed and covered
137 by regression assertions before publication.
138 
139 The 8,038 exact API checks compare separate constructions: Picard integration,
140 sine/cosine tangent division, root filters in a cyclotomic quotient,
141 direct and phase-matrix taps, and coefficient-level antiderivatives.
142 They verify finite implementation ranges. The algebraic laws above have
143 separate proofs in Appendix E.
144 
145 The catalog is ready for comparison with OEIS, with known anchors identified.
146 It makes no claim that an unmatched prefix establishes a new sequence, and
147 no sequence is represented here as a submitted or independently assigned OEIS entry.

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