### 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 |

Publication status | Published - 2000 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Aequationes Mathematicae*,

*59*(1-2), 1-19.

**On the Hurwitz-Lerch zeta-function.** / Kanemitsu, Shigeru; Katsurada, Masanori; Yoshimoto, Masami.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 59, no. 1-2, pp. 1-19.

}

TY - JOUR

T1 - On the Hurwitz-Lerch zeta-function

AU - Kanemitsu, Shigeru

AU - Katsurada, Masanori

AU - Yoshimoto, Masami

PY - 2000

Y1 - 2000

N2 - 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(e2πiξ, 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).

AB - 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(e2πiξ, 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).

KW - Hurwitz zeta-function

KW - Lerch zeta-function

KW - Power series expansion

KW - Special values of zeta-functions

UR - http://www.scopus.com/inward/record.url?scp=0000069849&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000069849&partnerID=8YFLogxK

M3 - Article

VL - 59

SP - 1

EP - 19

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 1-2

ER -