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

Abstract

This note is concerned with series of the form ∑ f(aⁿ) and ∑ 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 integral evaluations involving Riemann's zeta function ζ(s), limits, and series representations containing hyperpowers. Several examples of such sums are provided, each examined for possible new structure. In one case, a generalization of Riemann's classic relationship among the zeta, gamma, and Jacobi theta functions is obtained.