具有多项式指数的沃利斯型乘积与负整数处的狄利克雷β函数
Wallis-type products with polynomial exponents and the Dirichlet beta function at negative integers
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中文总结 AI 辅助
本研究提出构造带多项式指数的有理块无穷乘积的方法，推导多重伽马值相关的显式乘积，并证明带广义欧拉权重的有限多重伽马模板可对负整数处狄利克雷β函数的Duke-Imamoğlu表达式求值，得到奇数情形的收敛沃利斯-欧拉乘积。
中文摘要 AI 辅助
我们提出了一种设计有理块无穷乘积的方法，这些有理块的指数是下标$k$的多项式。通过匹配槽位常数的前$n$阶幂和，可迫使带有二项式指数$\binom{n+k-2}{n-1}$的$N$型乘积收敛于Vignéras多重伽马值$\boldsymbol{\textit{Γ}}_n$的比值；一个类比查找器可将任意1型求值结果推广到所有更高类型，随后二项式指数的整数组合可实现任意整值多项式指数，从而得到指数为$k$、$k^2$、$k^3$……的显式乘积，对应$\boldsymbol{\textit{π}}/2$、$\boldsymbol{\textit{√2}}$、$\boldsymbol{\textit{e}}^{2\boldsymbol{\textit{K}}/π}$以及$\boldsymbol{\textit{π}}^{M!}$的有理倍数等常数（第一部分）。作为主要应用（第二部分），我们证明对每个正整数$n$，带有广义欧拉权重$T(n,k)$（OEIS A225118）的有限多重伽马模板$\boldsymbol{\textit{𝒮}}_n$可对Duke-Imamoğlu表达式$\boldsymbol{\textit{𝒟}}_n = β'(-n) + (\boldsymbol{\textit{log 4}})\boldsymbol{\textit{β}}(-n)$求值。对奇数$n$，这给出了$\boldsymbol{\textit{e}}^{β'(-n)}$的收敛沃利斯-欧拉乘积；对偶数$n$，原始乘积发散。证明过程通过多重伽马函数方程展开模板，以闭式形式求出四分之一整数系数，并将得到的欧拉-二项式和与Duke多项式$P_{n+1,\boldsymbol{\textit{ℓ}}}$对应起来。
英文摘要
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 $\binom{n+k-2}{n-1}$ 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 $π/2$, $\sqrt{2}$, $e^{2K/π}$, 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 $\mathcal{S}_n$ with generalised Eulerian weights $T(n,k)$ (OEIS A225118) evaluates the Duke-Imamoğlu expression $\mathcal{D}_n = β'(-n) + (\log 4)\,β(-n)$. 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 $P_{n+1,\ell}$.
