The Zeta Function of a Recurrence Sequence of Arbitrary Degree
[EN]We consider a Dirichlet series $\sum_{n=1}^{\infty} a_{n}^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex plane, giving explicit formulas for its pole set an...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad de Salamanca (USAL) |
| Repositorio: | GREDOS. Repositorio Institucional de la Universidad de Salamanca |
| OAI Identifier: | oai:gredos.usal.es:10366/168501 |
| Acceso en línea: | http://hdl.handle.net/10366/168501 |
| Access Level: | acceso abierto |
| Palabra clave: | Linear recurrence sequence Dirichlet series Analytic continuation zeta function 12 Matemáticas |
| Sumario: | [EN]We consider a Dirichlet series $\sum_{n=1}^{\infty} a_{n}^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex plane, giving explicit formulas for its pole set and residues, as well as for its finite values at negative integers, which are shown to be rational numbers. To illustrate the results, we focus on some concrete examples which have also been studied previously by other authors. |
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