Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.

MATHIEU-TYPE SERIES BUILT BY (p, q)-EXTENDED GAUSSIAN HYPERGEOMETRIC FUNCTION (p, q)-extended Beta function;(p, q)-extended Gaussian hypergeometric function;integral representations;Mathieu-type series;Cahen formula;bounding inequality; The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and its associated alternating version whose terms contain a (p, q)-extended Gauss' hypergeometric function. Certain upper bounds for the two series are also given.

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.

INCOMPLETE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROPERTIES gamma functions;incomplete gamma functions;Pochhammer symbol;incomplete Pochhammer symbols;incomplete generalized hypergeometric functions;incomplete Fox-Wright function;generalized Hurwitz-Lerch zeta function;incomplete Hurwitz-Lerch zeta function;fractional derivative operator;probability density function; Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.