×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.4153/CJM-1954-058-2 · Zbl 0058.10702
[2] DOI: 10.1093/imamat/33.2.109 · Zbl 0565.30034
[3] Campos L., Portugal Math. 43 pp 347– (1985)
[4] DOI: 10.1093/imamat/36.2.191 · Zbl 0607.33001
[5] Erdelyi A., Some applications of fractional integration (1963)
[6] DOI: 10.1137/0112002 · Zbl 0178.14401
[7] DOI: 10.1137/0113014 · Zbl 0158.12504
[8] Gorenflo R., Abel Integral Equation: Analysis and Applications (1991) · Zbl 0717.45002
[9] Grunwald A., Z. Math. Phys. 12 pp 441– (1867)
[10] DOI: 10.1137/0112051 · Zbl 0187.36901
[11] Higgins T., The use of fractional integral operators for solving nonhomogenous differential equations (1967)
[12] Kalla S., Int. J. Math. Stat. Sci. 2 pp 187– (1993)
[13] DOI: 10.1137/1018042 · Zbl 0324.44002
[14] Lavoie J. L., Fundamental Properties of Fractional Derivatives Via Pochhammer Integrals (1976) · Zbl 0322.26005
[15] Letnikov A., Mat. Sbornik 3 pp 1– (1868)
[16] Liouville J., J. de l’École Polytech. 13 pp 71– (1832)
[17] DOI: 10.1016/0925-8388(94)90560-6
[18] Nekrassov P., Mat. Sbornik 14 pp 45– (1888)
[19] K. Nishimoto,Fractional Calculus, Vols. 1–5, Descartes Press, Koriyama, 1984, 1987, 1989, 1991, 1996. · Zbl 0605.26006
[20] Oldham K., The Fractional Calculus (1974) · Zbl 0292.26011
[21] DOI: 10.1137/0118059 · Zbl 0201.44102
[22] Osler T., Leibniz rule, the chain rule and Taylor’s theorem for fractional derivatives (1970) · Zbl 0201.44102
[23] DOI: 10.1137/0502004 · Zbl 0215.12101
[24] DOI: 10.1137/0503001 · Zbl 0233.26006
[25] Osler T., Math. Comput. 26 pp 903– (1972)
[26] DOI: 10.1007/BF02395016 · Zbl 0033.27601
[27] Ross B., Fractional Calculus and Its Applications (1974)
[28] Samko S., Fractional Integrals and Derivatives: Theory and Applications (1993)
[29] Slater L., Generalized Hypergeometric Functions (1966) · Zbl 0135.28101
[30] DOI: 10.1080/10652469308819014 · Zbl 0822.33006
[31] Tremblay R., Une contribution à la théorie de la dérivée fractionnaire (1974)
[32] DOI: 10.1137/0510087 · Zbl 0418.26005
[33] DOI: 10.1016/j.amc.2006.09.076 · Zbl 1116.33001
[34] R. Tremblay, S. Gaboury, and B. Fugère,Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives, Integral Transform Spec. Funct., in press. DOI:10.1080/10652469.2012.665910. · Zbl 1270.26007
[35] Watanabe Y., Tôhoku Math. J. 34 pp 8– (1931)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.