Abstract
We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.
Highlights
The Hurwitz-Lerch Zeta function Φ(z, s, a) which is one of the fundamentally important higher transcendental functions is defined by Φ (z, s, a) := ∞ ∑ n=0(n zn + a)s (a ∈ C \ Z−0 ; s ∈ C when |z| < 1; (1)R (s) > 1 when |z| = 1) .The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function ζ(s), the Hurwitz zeta function ζ(s, a), and the Lerch zeta function ls(ξ) defined by ζ (s) n=1 1 ns
By integrating by part m times, we obtain. This allows us to modify the restriction R(α) < 0 to R(α) < m. Another representation for the fractional derivative is based on the Cauchy integral formula
We recall six fundamental theorems related to fractional calculus that will play central roles in our work
Summary
The Hurwitz-Lerch Zeta function Φ(z, s, a) which is one of the fundamentally important higher transcendental functions is defined by (see, e.g., [1, p. 121 et seq.]; see [2] and [3, p. 194 et seq.]). Motivated by the works of Goyal and Laddha [4], Lin and Srivastava [5], Garg et al [6], and other authors, Srivastava et al [7] (see [8]) investigated various properties of a natural multiparameter extension and generalization of the Hurwitz-Lerch zeta function Φ(z, s, a) defined by (5) (see [9]) In terms of the extended Hurwitz-Lerch zeta function defined by (6), the following generalization of several known integral representations arising from (5) was given by Srivastava et al [7] as follows:. (R (a) > 0; R (s + λα) > −λ) , provided that each member of (28) exists and (29)
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