Some families of the Hurwitz–Lerch Zeta functions and associated fractional derivative and other integral representations
Section snippets
Introduction and definitions
For the Hurwitz (or generalized) Zeta function ζ(s,a) defined, when , by (cf., e.g., [6, p. 88 et seq.])and continued meromorphically to the whole complex s-plane (except for a simple pole at s=1 with its residue 1), Yen et al. [9, p. 100, Theorem] derived the following sum–integral representation:A special case of the sum–integral representation (2) when k=2 was
A further generalization of the sum–integral representation (7)
We begin by recalling the Fox–Wright extension of the generalized hypergeometric function with p numerator and q denominator parameters, defined bywhere the argument z, the complex parameters αj (j=1,…,p) and βj (j=1,…,q), and the positive real parameters Aj (j=1,…,p) and Bj (j=1,…,q) are so constrained that the series in (12) converges absolutely (see, for details, [1, p. 183]). Since
Other miscellaneous representations
(I) First of all, we recall the Riemann–Liouville fractional derivative operator Dzμ defined by (cf. [2, p. 181 et seq.]; see also [5])Clearly, we havewhich, in view of the definition (8), yields the following fractional derivative formula for the generalized Hurwitz–Lerch Zeta function Φμ,ν(ρ,σ)(z,s,a) with ρ=σ:In
Acknowledgements
The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC 92-2912-I-033-004-A2, the Faculty Research Program of Chung Yuan Christian University, and the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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