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arXiv · 2309.15190

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

Larry Glasser·Michael Milgram
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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 $ζ(s)$, 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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Larry Glasser, Michael Milgram. 2024-09-18. On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences. https://arxiv.org/abs/2309.15190

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