Weighted hypergeometric functions and fractional derivative

Abstract

ABSTRACT: We introduce some weighted hypergeometric functions and the suitable generalization of the aputo fractional derivation. For these hypergeometric functions, some linear and bilinear relations are obtained by means of the mentioned derivation operator. Then some of the considered hypergeometric functions are determined in terms of the generalized Mittag-Leffler function E (γj),(lj) (ρj),λ [z1, ... ,zr] (Mainardi in Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010) and the generalized polynomials Sm n [x] (Srivastava in Indian J. Math. 14:1-6, 1972). The boundary behavior of some other class of weighted hypergeometric functions is described in terms of Frostman’s α-capacity. Finally, an application is given using our fractional operator in the problem of fractional calculus of variations.