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 $\binom{n+k-2}{n-1}$ to converge to a ratio of Vign\'eras multiple gamma values $\Gamma_n$; 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 $\pi/2$, $\sqrt{2}$, $e^{2K/\pi}$, and rational multiples of $\pi^{M!}$ (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-Imamo\u{g}lu expression $\mathcal{D}_n = \beta'(-n) + (\log 4)\,\beta(-n)$. For odd $n$ this yields a convergent Wallis-Eulerian product for $e^{\beta'(-n)}$; 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 $P_{n+1,\ell}$.