On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917778799198208 |
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| author | Glasser, Larry Milgram, Michael |
| author_facet | Glasser, Larry Milgram, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15190 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences Glasser, Larry Milgram, Michael Classical Analysis and ODEs 44A05, 44A20, 33B99, 40-08 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. |
| title | On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences |
| topic | Classical Analysis and ODEs 44A05, 44A20, 33B99, 40-08 |
| url | https://arxiv.org/abs/2309.15190 |