Extensions of Euler Type Sums and Ramanujan Type Sums

TL;DR

This paper introduces a new digamma function, extends Euler and Ramanujan sum identities, and generalizes tools for analyzing these sums, leading to new formulas and illustrative examples.

Contribution

It defines a new digamma function, extends existing tools for Euler sums, and generalizes Ramanujan-type identities involving hyperbolic series.

Findings

Immediate corollaries of Flajolet and Salvy's main results

New parameterized Ramanujan-type identities

Examples illustrating the extended formulas

Abstract

We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper \cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.