A new Leibniz rule and its integral analogue for fractional derivatives

In 1972, T.J. Osler proposed a generalization of the Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function. This new rule was based on one of his own result on Taylor's series for the fractional derivative he obtained in 1971. Later, he gave the integral analogue of that new Leibniz rule. In this paper, we present a new Leibniz rule for the fractional derivatives of the product of two functions with respect to an arbitrary function and we give its integral analogue. 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.

[1]  Thomas J. Osler,et al.  The Integral Analog of the Leibniz Rule , 1972 .

[2]  M. Riesz.  L'intégrale de Riemann-Liouville et le problème de Cauchy , 1949 .

[3]  Thomas J. Osler,et al.  Fractional Derivatives and Special Functions , 1976 .

[4]  R. Tremblay,et al.  Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives , 2013 .

[5]  Richard Tremblay.  Some Operational Formulas Involving the Operators $xD$, $x\Delta $ and Fractional Derivatives , 1979 .

[6]  J. H. Barrett,et al.  Differential Equations of Non-Integer Order , 1954, Canadian Journal of Mathematics.

[7]  L. Campos.  On rules of derivation with complex order for analytic and branched functions , 1986 .

[8]  T. Osler.  Taylor’s Series Generalized for Fractional Derivatives and Applications , 1971 .

[9]  Thomas J. Osler,et al.  Fundamental properties of fractional derivatives via pochhammer integrals , 1975 .

[10]  Theodore P. Higgins.  A Hypergeometric Function Transform , 1964 .

[11]  Richard Tremblay,et al.  The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions , 2007, Appl. Math. Comput..

[12]  Thomas J. Osler,et al.  Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series , 1970 .

[13]  B. Dwork.  Generalized Hypergeometric Functions , 1990 .

[14]  J. Hammond.  On General Differentiation , 1880 .

[15]  Thomas J. Osler,et al.  A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval’s Formula , 1972 .

[16]  Arthur Erdélyi.  Axially Symmetric Potentials and Fractional Integration , 1965 .

[17]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[18]  Hari M. Srivastava,et al.  The convolution method for the development of new leibniz rules involving fractional derivatives and of their integral analogues , 1993 .

[19]  L. M. B. C. Campos,et al.  On a Concept of Derivative of Complex Order with Applications to Special Functions , 1984 .

[20]  Francesco Mainardi,et al.  Fractional relaxation in anelastic solids , 1994 .

[21]  A. Erdélyi,et al.  An Integral Equation Involving Legendre Functions , 1964 .

[22]  Jürgen Kurths,et al.  Fractional calculus and its applications , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  L. M. B. C. Campos,et al.  On a Systematic Approach to some Properties of Special Functions , 1986 .

[24]  A. Erdélyi,et al.  SOME APPLICATIONS OF FRACTIONAL INTEGRATION , 1963 .

[25]  T. Osler.  Leibniz rule, the chain rule, and taylor's theorme for fractional derivatives , 1971 .