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Special-function machinery

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komorebi_math.special_functions provides numerical Gamma shape transforms and exact rational residue and scalar functions. It imports without loading mpmath, SciPy, or SymPy. Numerical evaluation needs the existing scientific extra; the exact rational functions need no additional dependency.

from fractions import Fraction
from komorebi_math.special_functions import (
    bernoulli_packet_integer,
    gamma_sech_ambiguity,
    beta_binomial_refinement,
    meixner_pollaczek,
    q_racah_scalar,
)

# Exact rational Bernoulli polynomial packet, with B_1(x)=x-1/2.
packet = bernoulli_packet_integer(4, Fraction(3, 7), 3, 1)

# Unit log-Gamma windows with two positive shapes.
overlap = gamma_sech_ambiguity("1.25", "2.75", ".8", ".4", dps=60)

# p_(n,alpha+gamma)(x+y) expands into these normalized child amplitudes.
coefficients = beta_binomial_refinement(4, "1.25", "2.75", dps=60)
polynomial = meixner_pollaczek(4, 4, ".6", dps=60)

# A terminating scalar value with explicit exact rational parameters.
scalar = q_racah_scalar(2, 3, "2/7", "3/5", "4/9", "5/8", "2/3")

Exact rational functions

bernoulli_packet_integer(n,a,M,r) evaluates M**(n-1)*B_n((a+r)/M) with rational arithmetic. It accepts any rational translation, including zero and negative values. It reuses the existing Bernoulli-number implementation rather than adding a second number sequence. Summing the M residues reconstructs B_n(a); parent packets also reconstruct from their child residues. Degree and residue must be nonnegative integers, M>=1, and r<M.

q_pochhammer, q_racah_lattice, q_racah_scalar, and racah_scalar accept integers, Fraction values, and exact rational strings. They reject floating inputs. The q functions require 0<q<1. The scalar sums retain terms through min(n,y) and reject a zero denominator in every retained term, before any cancellation. No value is assigned to a singular retained term.

These functions evaluate scalar formulas. They do not assert positive orthogonality or identify those formulas with the finite matrix recoupling construction. An exact counterexample illustrates the distinction: for q=1/2, alpha=beta=1/2, gamma=4, delta=1, and n,y in {0,1}, both lattice sites equal 3 and every table entry equals 1. The termination condition gamma*q=q**(-1) therefore does not establish a positive orthogonal family.

Numerical families and conventions

Numerical functions return mpmath scalar values. Each call creates an isolated precision context; dps is the requested decimal precision and must be at least 15. The process-wide mpmath precision is unchanged. Shapes must be positive finite real values. Real arguments must be finite, and booleans are excluded from all scalar and integer domains.

Shape functions use extra isolated working digits to resolve cancellation at large parameters, then round the returned result to the requested precision. More than 10,000 working decimal digits, or excessive positive window-exponent work, raises ResourceLimitError. This work limit is not a uniform bound on elapsed time, and extra precision does not make the output a directed enclosure.

Functions Object and convention
log_gamma_window, log_gamma_fourier Unit window g_s(y)=Gamma(s)**(-1/2)*exp(s*y/2-exp(y)/2); unitary Fourier transform with exp(-i*omega*y)
gamma_sech_ambiguity, shape_affinity Centered overlap integral g_alpha(y+h/2)*g_gamma(y-h/2)*exp(i*eta*y) dy; zero-shift affinity
meixner_density, meixner_characteristic Fourier energy density and characteristic function sech(h/2)**s
beta_logit_density Density of log(U/(1-U)) for U~Beta(alpha,gamma)
meixner_pollaczek Normalized polynomial with generating function (1-i*z)**(-s/2+i*x)*(1+i*z)**(-s/2-i*x) before normalization
beta_binomial_refinement, shape_refinement_coefficient Positive two-child and many-child refinement amplitudes; degree vectors and shape vectors must have the same positive length
shape_transfer_singular_value sqrt((s)_n/(s+t)_n) for s>0 and increment t>0
bernoulli_hurwitz, bernoulli_hurwitz_packet Entire-order -z*zeta(1-z,a) and residue refinement for real a>0; exact removable values at z=0

The Bernoulli evaluator retains tiny nonzero orders instead of substituting the zero-order value. It temporarily increases its isolated working precision to resolve the Hurwitz pole cancellation, with a 10000-digit working cap. The integer-order polynomial function has a broader translation domain than the complex-order Hurwitz function.

These are numerical evaluations, not interval enclosures. dps alone does not certify an error bound. Integration tests compare independent quadrature, generating-function coefficients, exact residue reconstruction, polynomial addition, and refinement-tree composition in a stated finite range. They do not certify arithmetic depth asymptotics, unresolved signed estimates, or all proofs in the source manuscripts.

The coordinates, analytic, generating, and sequences namespaces provide residue packets, finite analysis and synthesis, normalized generating functions, and classical polynomials. This namespace supplies the Gamma/Meixner shape family, Bernoulli–Hurwitz order interpolation, and a finite scalar q-Racah bridge. Infinite arithmetic transfer estimates, a full representation-theoretic matrix recoupling engine, and a continuous-series spectral transform are outside its implemented contract.

Exact counterexamples and independent numerical comparisons support the stated conventions. Bounded numerical tests do not establish universal identities or replace mathematical proofs.