## A new Leibniz rule and its integral analogue for fractional derivatives.(English)Zbl 1267.26006

T. Osler has done a lot of work on the Leibniz rule for fractional derivatives of the product of two functions with respect to an arbitrary function [SIAM J. Appl. Math. 18, 658–674 (1970; Zbl 0201.44102); SIAM J. Math. Anal. 3, 1–16 (1972; Zbl 0233.26006); Math. Comput. 26, 903–915 (1972; Zbl 0256.26006)]. The reviewer, Y. F. Luchko and S. B. Yakubovich [Int. J. Math. Stat. Sci. 2, No. 2, 187–225 (1993; Zbl 0858.26005)] have investigated a new Leibniz rule by the method of G-convolutions. In this paper, the authors obtain another new generalized Leibniz rule for the fractional derivatives of the product of two functions and its integral analogue. The desired result is obtained by an appeal to a recent result of the authors [Integral Transforms Spec. Funct. 24, No. 1, 50–64 (2013; Zbl 1270.26007)], concerning power series expansion of an analytic function in terms of a rational function. The advantage of this method is that it considerably decreases the number of restrictions on the different parameters. Further, any analytic function, even those with an essential singularity at the origin, can be considered. Finally, new series expansions and definite integrals involving special functions are derived as special cases of the new Leibniz rule and the corresponding integral analogue.

### MSC:

 26A33 Fractional derivatives and integrals 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33C52 Orthogonal polynomials and functions associated with root systems
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### References:

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