Probabilistic proof of a summation formula

TL;DR

This paper introduces a probabilistic approach using hyperbolic secant random variables to derive a summation formula for an alternating series and an expression for the zeta function, involving Euler numbers.

Contribution

It presents a novel probabilistic method to derive identities related to the zeta function and Euler numbers, expanding the tools available for such derivations.

Findings

Derived a summation formula for an alternating infinite series.

Obtained an expression for the zeta function involving Euler numbers.

Connected moments of hyperbolic secant variables to these identities.

Abstract

The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function 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.