Abstract
In the present paper, we aim to extend the Hurwitz–Lerch zeta function varPhi _{delta ,varsigma ;gamma }(xi ,s,upsilon ;p) involving the extension of the beta function (Choi et al. in Honam Math. J. 36(2):357–385, 2014). We also study the basic properties of this extended Hurwitz–Lerch zeta function which comprises various integral formulas, a derivative formula, the Mellin transform, and the generating relation. The fractional kinetic equation for an extended Hurwitz–Lerch zeta function is also obtained from an application point of view. Furthermore, we obtain certain interesting relations in the form of particular cases.
Highlights
In the present paper, we aim to extend the Hurwitz–Lerch zeta function Φδ,ς;γ (ξ, s, υ; p) involving the extension of the beta function
It is clearly seen that Eqs. (1.10) and (1.11) are particular cases of Eq (1.12) and Eq (1.13), respectively, provided when p = q. Motivated by those various fascinating extensions of Hurwitz–Lerch zeta function, further we establish an extension of generalized Hurwitz–Lerch zeta function involving extended beta function B(δ1, δ2; p, q)
(iii) On taking the values of δ = γ = 1 in Eq (2.1), we find a new particular case of the extended generalized zeta function Φδ∗(ξ, s, υ) established by Goyal and Laddha [9]: Φς∗(ξ, s, υ; p, q) = Φ1,ς,1(ξ, s, υ; p, q) =
Summary
We aim to extend the Hurwitz–Lerch zeta function Φδ,ς;γ (ξ , s, υ; p) involving the extension of the beta function We study the basic properties of this extended Hurwitz–Lerch zeta function which comprises various integral formulas, a derivative formula, the Mellin transform, and the generating relation. The fractional kinetic equation for an extended Hurwitz–Lerch zeta function is obtained from an application point of view. In 2014 Parmar and Raina [15] introduced the generalized Hurwitz–Lerch zeta function involving the extended beta function [16] given by. (v) The limiting case of new extension of the generalized Hurwitz–Lerch zeta function involving extended beta function Φς∗;γ (ξ , s, υ; p, q) is given by. Employing the above result in Eq (2.1) and interchanging the order of summation and integration (condition above), we obtain ts–1e–υt (δ)m
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