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A generalization of the Hurwitz-Lerch zeta function. (English) Zbl 1145.11068

The generalization in the title is defined as the analytic continuation of the series

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

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 ${{\Phi }}_{n}\left(z,s,a-t\right)$ as a power series in $t$ whose $k$th coefficient involves ${{\Phi }}_{n}\left(z,s+k,a\right)$. This basic identity leads to interesting evaluations of classes of series associated with ${{\Phi }}_{n}\left(z,s,a\right)$.

##### 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.