An expression for zeta values and a summation formula via hyperbolic secant random variables

Year 2025, Volume: 54 Issue: 5, 1897 - 1904, 29.10.2025

Tae-kyun Kım , Dae San Kim

https://doi.org/10.15672/hujms.1592384

Abstract

The aim of this paper is to derive a summation formula for the series $\sum_{k=0}^{\infty} \frac{(-1)^{k}} {(2k+1)^{2n+1}}$ and an expression for $\zeta(2n+2)$ by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing the moments of the random variable and the moments of the sum of two independent such random variables.

Keywords

Euler numbers; zeta function , hyperbolic secant random variable , moment generating function; probability density function

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