### Abstract

Let Φ(z, s, α) = Σ^{∞}_{n=0} z ^{n}/(n+α)^{s} be the Hurwitz-Lerch zeta-function and φ(ξ, s, α) = Φ(e^{2πiξ}, s, α) for ξ ∈ ℝ its uniformization. Φ(z, s, α) reduces to the usual Hurwitz zeta-function ζ(s, α) when z = 1, and in particular ζ(s) = ζ(s, 1) is the Riemann zeta-function. The aim of this paper is to establish the analytic continuation of Φ(z, s, α) in three variables z, s, α (Theorems 1 and 1*), and then to derive the power series expansions for Φ(z, s, α) in terms of the first and third variables (Corollaries 1* and 2*). As applications of our main results, we evaluate in closed form a certain power series associated with ζ(s, α) (Theorem 5) and the special values of φ(ξ, s, α) at s = 0, -1, -2,... (Theorem 6).

Original language | English |
---|---|

Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Aequationes Mathematicae |

Volume | 59 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

### Fingerprint

### Keywords

- Hurwitz zeta-function
- Lerch zeta-function
- Power series expansion
- Special values of zeta-functions

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Aequationes Mathematicae*,

*59*(1-2), 1-19. https://doi.org/10.1007/PL00000117