Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

TL;DR

This paper introduces a new class of zeta series transformations linked to generalized Stirling numbers, enabling novel series expansions for special functions and constants, with applications to Dirichlet beta, Legendre chi, and zeta functions.

Contribution

It develops a generalized transformation framework for zeta series that connects to harmonic numbers and Stirling numbers, providing new tools for series expansions of special functions.

Findings

Derived new series expansions for the Dirichlet beta and Legendre chi functions.

Established BBP-type series identities for mathematical constants.

Presented novel series representations for zeta function values at integers.

Abstract

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers $s \geq 3$.