Exact sequence, polyphase and tap laboratory
This chapter connects the nonlinear family to exact sequences and finite filter algebra. Its reusable code implements finite calculations. The analytic results are proved in Chapter 1 and Appendix A. All mathematical sequence indices begin at zero. All finite arrays are zero extended beyond their displayed support. The catalog contains explicit candidates for OEIS lookup. This chapter does not assert an OEIS accession or historical priority for an unsearched row.
The integer polynomial object
Write the unique odd formal solution as
The two coefficient systems are related by \(A_n=(2n+1)!c_n\). In particular, \(A_n\) is the derivative of order \(2n+1\) at zero. The coefficient \(c_n\), the derivative \(A_n\), and the full EGF sequence \(a_m\) with \(a_{2n+1}=A_n, a_{2n}=0\) are different objects. None of the code silently substitutes one for another.
The nonlinear differential equation gives the recurrence
Proof.
Multiply the ordinary Taylor recurrence by \((2n+1)!\). The factor attached to \(A_iA_j\), where \(i+j=n-1\), is \((2n)!/[(2i+1)!(2j+1)!]=\binom{2n}{2i+1}\). This also proves the recurrence directly in the EGF convention. \(\square\)
Polynomial theorem. For every \(n\), \(A_n\in\mathbb Z[q,r]\), all its nonzero coefficients are positive integers, and \(A_n(q,r)=A_n(r,q)\). Every monomial \(q^u r^v\) satisfies
Proof.
The recurrence is an induction using positive integer binomial coefficients. Swapping \(q,r\) and replacing \(i\) by \(n-1-i\) leaves the sum unchanged. If child indices are \(i,j\), their exponent pairs receive the shifts \((2i+1,2j+1)\). The lower bounds follow from \(i+j+1=n\). The two parities are the parity of \(i+j+1\), also \(n\). For the degree bound, a recurrence term has total degree at most
since \(i(i+1)+j(j+1)\) is largest at \((i,j)=(0,n-1)\) or its swap. The base \(A_0=1\) satisfies every assertion. \(\square\)
The first four polynomials are
Consequently each integer pair \((q,r)\) gives an integer sequence for all \(n\), independently of whether its formal Taylor series converges. For example, the first values of \(A_n(1,2)\) are \(1,4,160,35200,38041600\).
Rational feedback banks also admit integer sequences
For rational channel tuples \((h_t,q_t,r_t)\), use
The preceding recurrence is summed over channels with the factor \(h_t\). In the universal polynomial over the weights and contractions, every term of \(A_n\) has weight degree exactly \(n\) and total contraction degree at most \(n(n+1)\). The weight statement follows from one new weight at each recurrence step and \(1+i+j=n\); the contraction bound uses the same induction.
Let \(H\) be a common denominator for all channel weights, and \(D\) a common denominator for all channel contractions. Then
This is a proved normalization, even with signed rational weights. It is usually not the smallest possible denominator clearing. The catalog records both \(A_n\) and \(B_n\), and the exact multiplier, so that a proposed sequence lookup can state its normalization honestly.
A sharp distinction between sequences and analytic germs
The entire-function theorem uses \(0<q,r<1\). Specializations with large positive integers produce valid formal and integer sequences but do not necessarily produce analytic functions near zero.
Zero-radius theorem.
Read proof · Permalink · Source line 122
If \(q>0\) and \(r>1\), the formal series for \(T_{q,r}\) has radius of convergence zero. The same holds after swapping the parameters.
Proof.
Every coefficient is positive. In the ordinary recurrence keep only the term \(i=0\). This gives
Since the denominator is at most \((2n+1)^n\),
The Cauchy–Hadamard criterion therefore gives radius zero. \(\square\)
More generally, a positive feedback bank with a channel satisfying \(h>0,q>0,r>1\) has the same obstruction, replacing \(q^n\) by \((hq)^n\). Positivity prevents cancellation. No conclusion of this form is asserted for a bank with signed weights.
| Parameters or bank | First five \(A_n\) | Mathematical status |
|---|---|---|
| \((1,1)\) | \(1\), \(2\), \(16\), \(272\), \(7936\) | Known tangent derivatives |
| \((0,1)\) | \(1\), \(0\), \(0\), \(0\), \(0\) | Known linear boundary |
| \((1,-1)\) | \(1\), \(-2\), \(16\), \(-272\), \(7936\) | Known hyperbolic tangent derivatives |
| \((1,2)\) | \(1\), \(4\), \(160\), \(35200\), \(38041600\) | Integer sequence, radius zero |
| \((1/2,1/2)\) | \(1\), \(1/2\), \(1/4\), \(1/8\), \(1/16\) | Known scaled hyperbolic sine |
| \((1/4,3/4)\) | \(1\), \(3/8\), \(45/256\), \(21465/262144\), \(39833775/1073741824\) | Entire asymmetric family, order one |
The complete catalog contains 18 rows, each through \(n=12\): three classical or degenerate integer anchors, eight positive integer formal specializations, four rational contractions, one entire edge case of infinite order, and two normalized feedback banks. Integer integrality and rational clearing are established by the preceding proofs, not merely inferred from the finite sample.
The uncontracted boundary is entire of infinite order
Boundary theorem. For \(0<q<1\), the normalized formal solution of
extends to an entire function of infinite order. The swapped boundary has the same property.
Proof of entire extension. The tangent-coefficient majorant gives an initial holomorphic germ. Suppose \(T\) is known on \(|z|<R\). The function \(T(qz)\) is known on \(|z|<R/q\). On this larger disk solve the linear ODE \(U'=1+T(qz)U\), \(U(0)=0\). Explicitly,
The integrals are path independent on the disk. Linear-ODE uniqueness shows \(U=T\) on the previous disk, and there \(T(qz)\) equals the newly extended function at \(qz\). Thus the same nonlinear equation holds on the enlarged disk. Iterating \(R\mapsto R/q\) exhausts the plane.
Proof of infinite order. Positive coefficients imply \(T(x)\ge x\) for \(x>0\) and \(M(x)=T(x)\). The logarithmic derivative obeys
Integrating from one, with \(T(1)\ge1\), gives \(T(x)\ge\exp(q(x^2-1)/2)\) for \(x\ge1\). For \(R\ge2/q\), integrate the same logarithmic derivative over \([R/2,R]\). Since \(T(R/2)\ge1\),
Therefore \(\log\log M(R)/\log R\to\infty\), which proves infinite order. \(\square\)
For nonnegative parameters this gives a complete analytic/formal phase chart:
| Parameter region | Normalized solution |
|---|---|
| \(0<q,r<1\) | Entire, finite order defined by \(q^\rho+r^\rho=1\) |
| One parameter equals one, the other lies in \((0,1)\) | Entire, infinite order |
| \(q=r=1\) | Tangent, meromorphic with its usual poles |
| \(q>0,r>0,\max(q,r)>1\) | Formal solution with radius zero |
| One parameter equals zero | Linear solution \(z\), regardless of the other scale |
The finite-order formula applies in the open contraction square. It does not assign a finite order to the edge, where its defining scale equation has no positive root.
Exact root-filter sections
For an EGF \(G(z)=\sum_{m\ge0}a_m z^m/m!\), an integer base \(b\ge1\), a residue \(0\le j<b\), and \(\omega=e^{2\pi i/b}\), define
This holds as a formal series and wherever the original series converges. Indeed the coefficient multiplier is the geometric sum \(b^{-1}\sum_{k=0}^{b-1}\omega^{k(m-j)}\), equal to one or zero according to the residue. It proves all the projector identities
The full-position section and the decimated lane are distinct arrays. A section keeps the original positions, inserting zeros outside the residue. A lane \(x_j[n]=a_{bn+j}\) changes the index to \(n\).
For an ordinary generating function \(X(z)=\sum_ma_m z^m\), define \(X_j(w)=\sum_na_{bn+j}w^n\). Then
For an EGF the corresponding exact reconstruction is
One must retain \((bn+j)!\). Replacing it by \(n!\) gives a different function. The root-filter section identity has no such factorial ambiguity.
Ordinary taps, EGF multiplication and differentiation
Let \(h_0,\ldots,h_{L-1}\) be a finite tap vector and let \(a_m=0\) for \(m<0\). Ordinary taps form
For OGFs this is multiplication by \(H(z)=\sum_th_tz^t\): \(B(z)=H(z)X(z)\). For EGFs it is a sum of repeated zero-constant antiderivatives:
Proof.
The EGF of the delayed sequence is \(\sum_{m\ge t}a_{m-t}z^m/m!=I^tG(z)\). Summing taps gives the formula. Conversely, \(G^{(t)}\) has coefficient sequence \(a_{m+t}\), so derivatives sample future coefficients and do not implement these past delays. \(\square\)
For two EGFs \(G_a,G_h\), ordinary function multiplication instead gives the Hurwitz product
This is a binomial convolution. For example, with \(a_m=1\) and taps \(h=(1,1)\), ordinary taps give \(b_2=2\), while the Hurwitz product gives \(b_2=3\). This explicit negative control catches a common normalization error.
Intrinsic nonlinear feedback taps are the parameters \((h_t,q_t,r_t)\) inside \(F'=1+\sum_th_tF(q_tz)F(r_tz)\). They change the function's defining equation and its derivative recurrence. Ordinary FIR taps are a linear transformation of an already defined coefficient sequence. Both are useful, but they operate on different objects. The intrinsic-bank growth theorem is proved in Appendix A.
The exact finite polyphase tap matrix
For an OGF sequence \(a_m\), define lanes \(x_l[n]=a_{bn+l}\) and \(X_l(w)=\sum_nx_l[n]w^n\). Apply finite ordinary taps and obtain \(y_j[n]=b_{bn+j}\). Then the exact matrix identity is
Every exponent is a nonnegative integer. It records the carry across phase boundaries, rather than erasing it.
Proof.
A term \(h_t a_{bn+j-t}\) is in phase \(l\) exactly when \(t+l\equiv j\pmod b\). Write \(d=(t+l-j)/b\). That term is \(h_t x_l[n-d]\), whose OGF is \(h_tw^dX_l(w)\) by zero extension. Summing over taps and phases gives the matrix. \(\square\)
For two phases and two taps,
The upper-right factor \(w\) is necessary: a delay from odd phase to even phase crosses one block. A diagonal-only phase operation cannot implement this filter. The code constructs the sparse matrix for any finite base and tap vector and checks that recombination gives the full direct convolution.
Reuse and evidence
From the package’s research/ directory, standard Python is enough:
python family_lab.py coefficients --q 1/4 --r 3/4 --n 8
python family_lab.py catalog --n 12 --output-dir examples
python checks/family_lab_checks.py
The reusable functions include asymmetric_odd_derivatives, feedback_odd_derivatives, asymmetric_polynomials, section, phase_lanes, merge_lanes, ordinary_taps, hurwitz_product, derivative_shift, integral_shift, polyphase_tap_matrix, and apply_polyphase_matrix. All core arithmetic is fractions.Fraction or integer arithmetic. The phase and tap API applies to any finite rational sequence, not just the special-function catalog.
The final completed run passed 8,038 exact assertions and is recorded in evidence/family_lab_checks.json. Its scope includes:
18 catalog rows through \(A_{12}\), independently compared with dense formal Picard iteration through degree 25.
Bivariate polynomials through \(A_8\), with 540 nonzero monomials checked for integer positivity, symmetry, parity and the degree bound.
Tangent coefficients independently obtained by division of sine and cosine, plus hyperbolic tangent, scaled hyperbolic sine and linear anchors.
Root-filter multipliers reduced exactly in cyclotomic fields for bases one through seven and indices zero through 31.
Direct FIR convolution compared with phase-matrix convolution for 26 input lengths, seven bases and four tap vectors, including signed rational taps.
The antiderivative interpretation of EGF taps and the factorial conversion of Hurwitz products, independently compared with ordinary Taylor products.
Negative controls for omitted phase carries, aliasing lanes by concatenation, silent truncation, derivative/delay confusion and Hurwitz/FIR confusion.
The bounded checks validate the implemented paths in the recorded ranges. The universal polynomial, degree, clearing, root-filter and phase-matrix statements follow from the proofs above. Entire continuation and growth order require the analytic proofs, not these finite checks. Historical identification of a catalog row requires a separate literature comparison.
evidence/family_lab_checks.json records software version, code hashes, exact arithmetic, bounds, category counts and run time. It reports no failed assertion in this run. Independent audit identified two API input-validation defects: a fractional residue was accepted by the section operator, and an empty phase matrix accepted base zero. Both are now rejected and have passing regression checks. These defects affected invalid inputs, not the displayed identities. A draft prose example initially transcribed \(A_3(1,2)\) and \(A_4(1,2)\) incorrectly; the exact catalog comparison exposed the mismatch and corrected them to \(35200\) and \(38041600\). The code result did not change. Larger numerical evidence is unnecessary for these algebraic identities; an independent mathematical audit should examine the reductions.