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

The family of generalized Hurwitz-Lerch zeta functions is defined in the following form

Φ λ,μ;ν (ρ,σ,κ) (z,s,a):= n=0 (λ) ρn (μ) σn (ν κn ·n!)z n (n+a) s ,

where λ,μ; a,ν( - {0}); ρ,σ,κ + ; κ-ρ-σ>-1 when s,z; κ-ρ-σ=-1 and s when |z|<ρ -ρ σ -σ κ κ ; κ-ρ-σ=-1 and Re(s+ν-λ-μ)>1 when |z|=ρ -ρ σ -σ κ κ .

The authors establish the Mellin-Barnes type integral representations, relations with 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.

11M35Hurwitz and Lerch zeta functions
33C60Hypergeometric integrals and functions defined by them
33C05Classical hypergeometric functions, 2 F 1