Abstract. In the present paper, we obtain a new formula for the generalized Weyl dif-ferintegral operator in a compact form avoiding the occurrence of infinite series and thusmaking it useful in applications. Our findings provide interesting generalizations and uni-fications of the results given by several authors and lying scattered in the literature. 1. IntroductionGeneralized differintegral operatorsWe shall define the generalized Weyl differintegral operator of a function f ( x )[10, p. 529,eq.(2.2)] (see also [5-8, 16]) as follows :Let α , β and γ be complex numbers. The generalized Weyl fractional integral(Re( α ) > 0) and derivative (Re( α ) 0) , ( 1) q d q dx q J + q; −q;x;∞ f ( x ) , ( Re ( α ) 0 , 0 <Re ( α )+ q 1 ,q = 1 , 2 , 3 , ) . (1.1)where F stands for the well known Gauss hypergeometric function.The operator J