In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell function, and the Lauricella functions of several variables are given. 1. Introduction The fractional derivative of arbitrary order , , is an extension of the familiar th derivative of the function with respect to to nonintegral values of and is denoted by . The aim of this concept is to generalize classical results of the th order derivative to fractional order. Most of the properties of the classical calculus have been expanded to fractional calculus, for instance, the composition rule [1], the Leibniz rule [2, 3], the chain rule [4], and Taylor’s and Laurent’s series [5–7]. Fractional calculus also provides tools that make it easier to deal with special functions of mathematical physics [8]. The most familiar representation for fractional derivative of order of is the Riemann-Liouville integral [9]; that is, which is valid for and where the integration is done along a straight line from to in the -plane. By integrating by part times, we obtain This allows modifying the restriction to [10]. In 1970, Osler [2] introduced a more general definition of the fractional derivative of a function with respect to another function based on Cauchy’s integral formula. Definition 1. Let be analytic in the simply connected region . Let be regular and univalent on and let be an interior or boundary point of . Assume also that for any simple closed contour in through . Then if is not a negative integer and is in , the fractional derivative of order of with respect to is defined by For nonintegral , the integrand has a branch line which begins at and passes through . The notation on this integral implies that the contour of integration starts at , encloses once in the positive sense, and returns to without cutting the branch line. With the use of that representation based on the Cauchy integral formula for the fractional derivatives, Osler gave a generalization of the following result [11, page 19] involving the derivative of order of the composite function : where In particular, he found the following formula [4]: where the notation means the fractional derivative of order of with respect to . Osler proved the generalized chain rule by applying the generalized Leibniz rule [2] for fractional derivatives to an important fundamental
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