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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

Φ n (z,s,a)= m 1 ,,m n =0 z m 1 ++m n (m 1 ++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 Φ n (z,s,a-t) as a power series in t whose kth coefficient involves Φ n (z,s+k,a). This basic identity leads to interesting evaluations of classes of series associated with Φ n (z,s,a).

11M99Analytic theory of zeta and L-functions
33B15Gamma, beta and polygamma functions
42A24Summability and absolute summability of Fourier and trigonometric series
11M35Hurwitz and Lerch zeta functions
11M36Selberg zeta functions and regularized determinants
11M41Other Dirichlet series and zeta functions
42A16Fourier coefficients, special Fourier series, etc.