Algebraic representatives of the ratios $ζ(2n+1)/π^{2n}$ and $β(2n)/π^{2n-1}$

Abstract

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $Ξ_n,Λ_n\in\mathbb{Q}[x]$ arising from integral representations of $β(2n)/π^{2n-1}$ and $ζ(2n+1)/π^{2n}$. 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 $β(2n)/π^{2n}$ and $ζ(2n+1)/π^{2n+1}.$

Theorems & Definitions (43)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5: Smallest zeros from endpoint ratios at a left endpoint $a$
• proof