The Lerch-Type Zeta Function of a Recurrence Sequence of Arbitrary Degree
[EN]We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $\{a_{n}\}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex $s$-plane. Thus we may assoc...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad de Salamanca (USAL) |
| Repositorio: | GREDOS. Repositorio Institucional de la Universidad de Salamanca |
| OAI Identifier: | oai:gredos.usal.es:10366/168519 |
| Acceso en línea: | http://hdl.handle.net/10366/168519 |
| Access Level: | acceso abierto |
| Palabra clave: | Linear recurrence sequence Hurwitz and Lerch zeta functions Dirichlet series Analytic continuation 12 Matemáticas |
| Sumario: | [EN]We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $\{a_{n}\}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex $s$-plane. Thus we may associate a Lerch-type zeta function $\varphi(z,s,x)$ to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees $2$ and $3$. Our method generalizes a formula of Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the poles and residues of $\varphi$, which turn out to be polynomials in $x$. In addition we study the dependence of $\varphi(z,s,x)$ on $x$ and $z$, and its properties as a function of three complex variables. |
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