On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences

TL;DR

This paper explores series involving functions with Mellin transforms, deriving integral representations that connect to special functions like the Riemann zeta, Gamma, and Theta functions, revealing new structures and generalizations.

Contribution

It introduces novel integral representations for series of the form f(a^n) and f(n^{-a}) using Mellin transforms, leading to new relationships among special functions.

Findings

Derived integral representations for series involving Mellin transforms.

Established new connections between series and special functions like zeta, Gamma, and Theta.

Generalized Riemann's classical relationships among special functions.

Abstract

This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several ways yielding a selection of interesting integral evaluations involving Riemann's function $\zeta(s)$, limits and series representations containing hyperpowers. A number of examples of such sums are provided, each of which is investigated for possible new structure. In one case, we obtain a generalization of Riemann's classic relationship among the Zeta, Gamma and Jacobi Theta functions.