Series involving rational, factorial and power functions

TL;DR

This paper presents new finite series expansions involving special functions like the Hurwitz-Lerch zeta function, derived through contour integration, differential equations, and algebraic methods, expanding the understanding of series with rational, factorial, and power functions.

Contribution

It introduces novel finite series representations of special functions, particularly the Hurwitz-Lerch zeta function, using advanced mathematical techniques.

Findings

New finite expansions involving quotient functions and Hurwitz-Lerch zeta function

Extended series previously known, derived via differential equations and algebraic methods

Series expressed in terms of special functions with contour integration

Abstract

This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given. These results represent a new form of expressing this special function as a finite series where contour integration is required for derivation. Extended series previously known and derived are extended using differential equations and algebraic methods.