Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers
Abstract
We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index k. Matching power sums of the slot constants through order n forces the Type-N product with binomial exponent binomn+k−2n−1 to converge to a ratio of Vignéras multiple gamma values Γn; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents k, k2, k3, ... for constants such as π/2, √2, e2K/π, and rational multiples of πM! (Part I). As the main application (Part II) we prove that for every positive integer n, a finite multiple-gamma template 𝒮n with generalised Eulerian weights T(n,k) (OEIS A225118) evaluates the Duke-Imamoğlu expression 𝒟n = β'(−n) + (log 4) β(−n). For odd n this yields a convergent Wallis-Eulerian product for eβ'(−n); for even n the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials Pn+1,ℓ.
Related
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BibTeX
@misc{farrell2026wallistype,
title = {{Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers}},
author = {Joshua W. E. Farrell},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29741},
eprint = {2608.29741},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29741},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers
AU - Joshua W. E. Farrell
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29741
UR - https://arxiv.org/abs/2608.29741
AB - We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index k. Matching power sums of the slot constants through order n forces the Type-N product with binomial exponent binomn+k−2n−1 to converge to a ratio of Vignéras multiple gamma values Γ_n; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents k, k^2, k^3, ... for constants such as π/2, √2, e^2K/π, and rational multiples of π^M! (Part I). As the main application (Part II) we prove that for every positive integer n, a finite multiple-gamma template 𝒮_n with generalised Eulerian weights T(n,k) (OEIS A225118) evaluates the Duke-Imamoğlu expression 𝒟_n = β'(−n) + (log 4) β(−n). For odd n this yields a convergent Wallis-Eulerian product for e^β'(−n); for even n the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials P_n+1,ℓ.
ER -