Generating Function Transformations Related to Polylogarithm Functions and the $k$-Order Harmonic Numbers

TL;DR

This paper introduces new generating function transformations linked to polylogarithms and harmonic numbers, leading to novel identities, series expansions, and recurrence relations with applications to zeta constants and Bernoulli polynomials.

Contribution

It defines a new class of transformations with generalized coefficients, revealing properties and identities that extend harmonic number and polylogarithm analyses.

Findings

New recurrence relations for harmonic numbers

Series expansions for polylogarithm and zeta functions

Applications to Bernoulli polynomials and zeta constants

Abstract

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the $k$-order harmonic number sequences. Other applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new "almost" linear recurrence relations between the integer-order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, $\zeta(k)$, at integers $k \geq 2$.