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

2023/09/26 by Larry Glasser, Glasser, Larry, Michael Milgram +1
Mathematics · Physics and Astronomy · #33B99 #40-08 #44A05 #44A20 #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2309.15190

openalex publication_date 2023/09/26 · openalex created_date 2023/09/30 · openalex updated_date 2026/07/28

Abstract

This note is concerned with series of the forms ∑ 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 interesting integral evaluations involving Riemann's function ζ(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.

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