Generalized hypergeometric expansion related to the Hurwitz zeta function

TL;DR

This paper explores hypergeometric series representations of the Hurwitz zeta function through Mellin transforms of fractional parts and log-sine functions, revealing new analytic properties and convergence rates.

Contribution

It introduces novel hypergeometric series equalities for the Hurwitz zeta function derived from Mellin transforms, enhancing understanding of its analytic continuation and functional equation.

Findings

Series converge at exponential rates.

Equalities capture meromorphic extension and functional equation.

Provides explicit convergence rate estimates.

Abstract

We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in the imaginary component is carried over to these equalities.