A generalization of the Hurwitz-Lerch zeta function.

*(English)*Zbl 1145.11068The generalization in the title is defined as the analytic continuation of the series

$${{\Phi}}_{n}(z,s,a)=\sum _{{m}_{1},\cdots ,{m}_{n}=0}^{\infty}\frac{{z}^{{m}_{1}+\cdots +{m}_{n}}}{{({m}_{1}+\cdots +{m}_{n}+a)}^{s}},$$

with appropriate restrictions on $z$, $s$, $a$. The case $z=1$ was considered by *E. W. Barnes* in [Cambr. Trans. 19, 374–425 (1904; JFM 35.0462.01) and ibid. 19, 322–355 (1904; JFM 35.0462.02)]. Integral representations are obtained, together with a basic summation formula that expresses ${{\Phi}}_{n}(z,s,a-t)$ as a power series in $t$ whose $k$th coefficient involves ${{\Phi}}_{n}(z,s+k,a)$. This basic identity leads to interesting evaluations of classes of series associated with ${{\Phi}}_{n}(z,s,a)$.

Reviewer: Tom M. Apostol (Pasadena)

##### MSC:

11M99 | Analytic theory of zeta and L-functions |

33B15 | Gamma, beta and polygamma functions |

42A24 | Summability and absolute summability of Fourier and trigonometric series |

11M35 | Hurwitz and Lerch zeta functions |

11M36 | Selberg zeta functions and regularized determinants |

11M41 | Other Dirichlet series and zeta functions |

42A16 | Fourier coefficients, special Fourier series, etc. |