Some families of the Hurwitz–Lerch Zeta functions and associated fractional derivative and other integral representations

https://doi.org/10.1016/S0096-3003(03)00746-XGet rights and content

Abstract

The main object of this paper is to present, in a unified manner, a number of fractional derivative and other integral representations for several general families of the Hurwitz–Lerch Zeta functions. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.

Section snippets

Introduction and definitions

For the Hurwitz (or generalized) Zeta function ζ(s,a) defined, when R(s)>1, by (cf., e.g., [6, p. 88 et seq.])ζ(s,a):=∑n=01(n+a)s(R(s)>1;a∈CZ0;Z0:={0,−1,−2,…})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:ζ(s,a)=∑j=0k−11Γ(s)0ts−1e−(a+j)t1−e−ktdt(k∈N:={1,2,3,…};R(s)>1;R(a)>0).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 pFq function with p numerator and q denominator parameters, defined bypΨq1;A1),…,(αp;Ap);1;B1),…,(βq;Bq);z:=∑n=0j=1pj)Ajnj=1qj)Bjnznn!,where 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(n+a)−s=1Γ(s)0ts−1e−(n+a)tdt(

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])Dzμ{f(z)}:=1Γ(−μ)z0(z−t)−μ−1f(t)dt(R(μ)<0),dmdzmDzμ−m{f(z)}(m−1≦R(μ)<m(m∈N)).Clearly, we haveDzμzλ=Γ(λ+1)Γ(λ−μ+1)zλ−μ(R(λ)>−1),which, 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:Dzμ−ν{zμ−1Φ(zσ,s,a)}=Γ(μ)Γ(ν)zν−1Φμ,ν(σ,σ)(zσ,s,a)(R(μ)>0;σ∈R+).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.

References (9)

  • A. Erdélyi et al.
    (1953)
  • A. Erdélyi et al.
    (1954)
  • S.P. Goyal et al.

    On the generalized Riemann Zeta functions and the generalized Lambert transform

    Gaṅita Sandesh

    (1997)
  • K. Nishimoto et al.

    Some integral forms for a generalized Zeta function

    J. Fract. Calc.

    (2002)
There are more references available in the full text version of this article.
View full text