Analytical properties of the Hurwitz–Lerch zeta function

In the present paper, we aim to extend the Hurwitz–Lerch zeta function Φδ,ς;γ(ξ,s,υ;p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varPhi _{\delta ,\varsigma ;\gamma }(\xi ,s,\upsilon ;p)$\end{document} 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.

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(δ 1 , δ 2 ; p, q).

Integral representations differential formula
The section deals with the integral representation of the new extension of the generalized Hurwitz-Lerch zeta function involving the extended beta function (2.1) as follows.
Proof We know that the Eulerian integral of the gamma function obeys the following identity [18]: Employing the above result in Eq. (2.1) and then interchanging the order of summation and integration (condition above), we obtain In view of the definition (1.12) and (1.13), we obtain the required result (3.1).
Proof The integral representation of the Pochhammer symbol (δ) m is defined as By making use of the above relation in (2.1) and interchanging the order of summation and integration which may be admissible subject to the condition of Theorem 3.5, we obtain Applying (2.6), we get the required integral representation. Subsequently, we establish the underlying derivative formula of (2.1).

Mellin transform of the Hurwitz-Lerch zeta function and their relation between H-function
The Mellin transform of a suitable integrable function f (κ) with index ϕ is defined, generally, by Employing the underlying well-known integral representation (see [1,p. 360,Eq. (1.14) which gives the desired result 4.1.
Remark 4.1 The generalized Hurwitz-Lerch zeta function can be easily written in terms of H-function as appears in the literature [19,20] (see also [7, p. 316, Eq. (3.2)]):   Theorem 4.2 For p 0, q 0, δ ∈ C and |t| < 1, the underlying generating function holds: Proof Consider the left-hand side of the assertion (4.7) of Theorem 4.2 be denoted by K 1 and in view of definition (2.1), we obtain Inverting the order of summation of the above equality and after a little simplification, we get Now, employing the binomial expansion and in view of definition (2.1), we obtain the assertion (4.7) of Theorem 4.2.

Theorem 4.3
Let p, q 0, δ ∈ C and |t| < |υ|; s = 1 then the generating functions of Φ δ,ς ,γ (ξ , s; υ; p, q) is given by Proof Using the definition (2.1) in the right-hand side of (4.10), we have In view of expansion (4.9) and some little simplification of the above second equality, we are thus led to the assertion (4.11).

Fractional kinetic equation
This section deals with the fractional kinetic equation (FKE) involving the new extended Hurwitz-Lerch zeta function (2.1). The FKE has great significance in the field of astrophysics and mathematical physics. The solutions of FKE has many applications in various fields such as renormalization of the non-stationary problem near the phase transition point [21], the theory of turbulence [22], diffusion in porous media [23], and kinetics in viscoelastic media [24], which has been published in the literature of special functions.
In 2000 Haubold and Mathai [25] derived a fascinating result between the rate of change of reaction, the destruction rate, and the production rate given by where N = N (t) is the rate of reaction, δ(N t ) =: δ is the rate of destruction, p = p(N ) is the rate of production and N t signifies the function defined by N t (t * ) = N (tt * ); t * > 0.
Under spatial fluctuations or homogeneities where the quantity N (t) is neglected we arrive at a particular case of Eq. (5.1), which is given by (see [25,26]) where 0 D -1 t denotes the standard fractional integral operator. A fractional generalization of standard kinetic Eq. (5.2) is investigated by Haubold and Mathai [25] as follows: where 0 D -ω t is the familiar Riemann-Liouville fractional integral operator (see [27]) defined as (5.5) and they obtained the solution of (5.5) as follows: Moreover, Saxena and Kalla [25] obtained the underlying fractional kinetic equation: where N (t) is the number density of a given species at time t, N 0 = N (0) is the number density of that species at time t = 0, c is a constant and f ∈ L(0, ∞). Applying the Laplace transform to (5.7) (see [28]), is given by the following relation: where the E α,β (ξ ) denotes the generalized Mittag-Leffler function [29] is given by Proof The Laplace transform of the Riemann-Liouville fractional integral operator is given by [30] L 0 D -ω t f (t); p = p -ω F(p), (5.12) where F(p) is defined in (5.9). Now, taking the Laplace transform of both sides of Eq. (5.11), we obtain which is the required result.

Concluding remarks
In the present paper, it seems to be of interest that the extensions of Hurwitz-Lerch zeta function so obtained are very general in nature and, by specific parameters, can yield the previously defined Hurwitz-Lerch zeta function which is shown in this paper. On that account, they become of great importance from an application perspective. For example,  -; -, (0, 1), (ς + m, 1); (0, 1), (γς, 1); p, q , (6.2) where G represents Meijer's G-function (see [31, p. 7 where F 2 [·] represents one of the four Appell series F j (j = 1, 2, 3, 4) (see [32, pp. 22-23]). Similarly, we can further obtain the connection with the Macdonald function and the Whittaker function.