Algebraic representatives of the ratios $\zeta(2n+1)/\pi^{2n}$ and $\beta(2n)/\pi^{2n-1}$

TL;DR

This paper provides explicit formulas and structural analysis of polynomials related to ratios of special functions, potentially aiding in understanding their arithmetic properties.

Contribution

It introduces explicit closed-form expressions for polynomials associated with these ratios, expanding on previous integral representations.

Findings

Explicit formulas for polynomials in terms of Eulerian numbers

Structural properties of these polynomials analyzed

Potential implications for the arithmetic nature of the ratios

Abstract

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $\Xi_n,\Lambda_n\in\mathbb{Q}[x]$ arising from integral representations of $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$. In this paper we give explicit closed formulae for these polynomials in terms of Eulerian numbers and study their structural properties. These properties may prove useful in studies on the arithmetic nature of the ratios $\beta(2n)/\pi^{2n}$ and $\zeta(2n+1)/\pi^{2n+1}.$