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Srivastava, H. M.; Saxena, Ram K.; Pogány, Tibor K.; Saxena, Ravi

Integral and computational representations of the extended Hurwitz-Lerch zeta function. (English) Zbl 1242.11065

Integral Transforms Spec. Funct. 22, No. 7, 487-506 (2011).
The family of generalized Hurwitz-Lerch zeta functions is defined in the following form \[ \Phi_{\lambda,\mu;\nu}^{(\rho,\sigma,\kappa)}(z,s,a):=\sum_{n=0}^{\infty}\frac{(\lambda)_{{\rho n}}(\mu)_{\sigma n}}{(\nu_{\kappa n}\cdot n!)}\frac{z^n}{(n+a)^s}, \] where \(\lambda,\mu \in \mathbb C\); \(a,\nu \in \mathbb C \setminus (\mathbb Z^- \cup\{0\})\); \(\rho, \sigma, \kappa \in \mathbb R^+\); \(\kappa-\rho-\sigma>-1\) when \(s,z \in \mathbb C\); \(\kappa-\rho-\sigma=-1\) and \(s\in \mathbb C\) when \(|z|<\rho^{-\rho}\sigma^{-\sigma}\kappa^{\kappa}\); \(\kappa-\rho-\sigma=-1\) and \(\text{Re}(s+\nu-\lambda-\mu)>1\) when \(|z|=\rho^{-\rho}\sigma^{-\sigma}\kappa^{\kappa}\).
The authors establish the Mellin-Barnes type integral representations, relations with \(\overline H\)-function, fractional derivatives and analytic continuation formulas which provides an extension of the analytic continuation formula for the Gauss hypergeometric function. Also, they present an extension of the generalized Hurwitz-Lerch zeta function, i.e, the special cases associated with the Mittag-Leffler type functions and the generalized \(M\)-series are obtained.
Reviewer: Roma Kačinskaitė (Siauliai)

Cited in 5 Reviews
Cited in 47 Documents

MSC:

11M35 Hurwitz and Lerch zeta functions
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C05 Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Riemann zeta function; Lerch zeta function; polylogarithmic function; general Hurwitz-Lerch zeta function; Gauss hypergeometric function; Fox-Wright \(\Psi\)-function; \(\overline H\)-function; Mittag-Leffler type functions; Mellin-Barnes type integral representations; analytic continuation

Citations:

JFM 37.0427.01
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\textit{H. M. Srivastava} et al., Integral Transforms Spec. Funct. 22, No. 7, 487--506 (2011; Zbl 1242.11065)
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