Gamma,Psi,Bernoulli Functions via Hurwitz Zeta Function

TL;DR

This paper provides new proofs and unified insights into classical special functions such as Bernoulli polynomials, gamma, and Psi functions using properties of the Hurwitz zeta function, including derivations of known formulas and relations.

Contribution

It introduces new natural proofs for classical results and unifies the derivation of properties of Bernoulli polynomials, gamma, and Psi functions via the Hurwitz zeta function.

Findings

New proofs of Bernoulli polynomial properties

Unified derivation of gamma and Psi function formulas

Confirmation of Stirling's approximation through zeta function relations

Abstract

Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified fashion.We also give a new proof of the relation between log gamma and derivative of the Hurwitz zeta function,including that of Stirling's expression for log gamma.