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Algebraic representatives of the ratios zeta(2n+1)/π^{2n} and β(2n)/π^{2n-1}
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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 $\zeta(2n+1)/\pi^{2n+1}.$
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Cited by 2 Pith papers
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Right edge rates of the zeros of $\widetilde{\Xi}_n$ and $\widetilde{\Lambda}_n$
The rightmost zeros of the rescaled polynomials approach 1 at exponential rates 4^{-(n-1)} and 9^{-(n-1)} respectively.
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Structure and Zero Asymptotics of Differential Operators Associated with ${\Xi}_n$ and ${\Lambda}_n$
Differential operators D_Ξ and D_Λ associated with polynomial families Ξ_n and Λ_n admit factorizations and hypergeometric descriptions, preserve hyperbolicity and zero positions, and yield iterated sequences whose ze...
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