Several classical identities via Mellin's transform

TL;DR

This paper introduces a Mellin transform-based summation rule that simplifies proofs of classical identities involving special functions, Bernoulli and Euler polynomials, and related zeta function values.

Contribution

It provides a novel Mellin transform approach to derive and prove classical identities more succinctly than traditional methods.

Findings

Expresses Hurwitz zeta function values at negative integers using Bernoulli polynomials

Derives identities involving exponential and Hermite polynomials

Offers a unified proof technique for classical special function relations

Abstract

We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.