Further generalization of the extended Hurwitz-Lerch Zeta functions

Abstract

Recently various extensions of Hurwitz-Lerch Zeta functions have been investigated. Here, we first introduce a further generalization of the extended Hurwitz-Lerch Zeta functions. Then we investigate certain interesting and (potentially) useful properties, systematically, of the generalization of the extended Hurwitz-Lerch Zeta functions, for example, various integral representations, Mellin transform, generating functions and extended fractional derivatives formulas associated with these extended generalized Hurwitz-Lerch Zeta functions. An application to probability distributions is further considered. Some interesting special cases of our main results are also pointed out.

A Double Integral Containing the Fresnel Integral Function S(x): Derivation and Computation

“…Proof. Use equation (18), set k � − 2, m � − (3/4), and simplify using equation (2.2.1.2.7) in [20].…”

Extended Wang sum and associated products

“…In the work of Choi et al [ 10 ], a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables was introduced. In the work of Parmar et al [ 11 ], further generalization of the extended Hurwitz-Lerch Zeta functions were developed. In the work of Parmar et al [ 12 ] a generalized form of the extended Hurwitz-Lerch Zeta function was considered with an emphasis on obtaining classical properties which include various integral representations, a differential formula, Mellin transforms, and certain generating relations.…”

New extensions of associated Laguerre polynomials

“…In many areas of applied mathematics, different types of special functions have became necessary tool for the scientists and engineers. During the recent decades or so, various interesting extensions of different special functions such as gamma and beta functions, the Gauss hypergeometric function, and so on have been introduced by several authors (see [1,4,5,[13][14][15]). In 1997, Chaudhry et al [4] have introduced the extension of Euler's Beta function as follows:…”

“…Recently, the extended Beta function B(x, y; p) and its systemic generalizations are used to introduce new extended special functions ( see [12,13,15]). In [13], Özarslan and Yilmaz introduced the extended Mittag-Leffler function E γ;c α,β (z; p) as follows:…”

“…Motivated by the works given in [12,13,15], we present new extensions for the associated Laguerre polynomials in terms of the extended Beta function B(x, y; p). For this aim, we recall that 2-variable associated Laguerre polynomials (2VALP) L n (x, y) are defined as [6]:…”