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Zbl 1145.11068
Choi, Junesang; Jang, Douk Soo; Srivastava, H.M.
A generalization of the Hurwitz-Lerch zeta function. (English)
[J] Integral Transforms Spec. Funct. 19, No. 1, 65-79 (2008). ISSN 1065-2469; ISSN 1476-8291/e

The generalization in the title is defined as the analytic continuation of the series $$\Phi_n(z,s,a)= \sum^\infty_{m_1,\dots, m_n= 0} {z^{m_1+\cdots+ m_n}\over (m_1+\cdots+ m_n+ a)^s},$$ with appropriate restrictions on $z$, $s$, $a$. The case $z= 1$ was considered by {\it 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 $\Phi_n(z,s,a)$.
[Tom M. Apostol (Pasadena)]

MSC 2000:: *11M99 Analytic theory of zeta and L-functions
33B15 Gamma-functions, etc.
42A24 Summability of trigonometric series
11M35 Other zeta functions
11M36 Selberg zeta functions and regularized determinants
11M41 Other Dirichlet series and zeta functions
42A16 Fourier coefficients, etc.

Keywords: gamma function; double gamma function; series associated with the zeta function; Riemann zeta fnction; Hurwitz zeta function; log-sine integrals; polylogarithms

Citations: JFM 35.0462.01; JFM 35.0462.02

Cited in: Zbl 1146.33001

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