On fractional calculus of the bivariate Mittag-Leffler function
On fractional calculus of the bivariate Mittag-Leffler function
Abstract
This article deals with a study of some fractional calculus properties of the bivari-ate Mittag-Leer functions including the fractional integrals and derivatives and
the singular integral equation involving the bivariate Mittag-Leer functions in the kernel. Further, we introduce a fractional integral operator involving bivariate
Mittag-Leer functions in the kernel. Also, we discuss the links of our ndings with known cases.
Downloads
Download data is not yet available.
References
[1] Berge C.Principles of combinatorics, New York: Academic Press; 1971.
[2] Bin-Saad Maged Gumman, Anvar Hasanov, Michael Ruzhansky. Some properties related
to the Mittag{Leer function of two variables. Integral Transforms and Special Functions.
https://doi.org/10.1080/10652469.2021.1939328.
[3] Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Transcendental Functions, Vol.
I, New York, Toronto and London: McGraw-Hill Book Company, 1953.
[4] Goren
o R, Mainardi R. On Mittag-Leer function in fractional evaluation processes. Journal
of Computational and Applied Mathematics 2000; 118: 283-299.
[5] Goren
o R, Mainardi F, Srivastava HM. Special functions in fractional relaxation oscillation
and fractional di usion-wave phenomena, In Proceedings of the Eighth International Colloquium
on Di erential Equations (Plovdiv, Bulgaria; August 18{23, 1997), ed. D. Bainov,
195{ 202. Utrecht: VSP Publishers.; 1998.
[6] Kilbas AA, Saigo M. On Mittag-Leer type function, fractional calculus operators and solution
of integral equations. Integral Transforms and Special Functions 1996; 4: 355-370.
[7] Kilbas AA , Saigo M, Saxena RK. Generalized Mittag-Leer function and generalized fractional
calculus operators. Integral Transforms and Special Functions 2004; 15: 31-49.
[8] Kilbas AA, Saigo M. On Mittag-Leer type function, fractional calculus operators and solutions
of integral equations. Integral Transforms and Special Functions 1996; 4: 355{370.
[9] Kilbas AA, Saigo M, Saxena RK. Generalized Mittag-Leer function and generalized fractional
calculus operators. Integral Transforms and Special Functions 2004; 15: 31{49.
[10] Mittag-Leer GM. Sur la nouvelle fonction E (x). Comptes rendus de l'Academie des Sciences
1903; 137: 554-558.
[11] Podlubny I. Fractional di erential equations, San Diego:Academic Press, 1999.
[12] Prabhakar TR . A singular integral equation with a generalized Mittag-Leer function in the
kernel. Yokohama Mathematical Journal 1971; 19 :7-15.
[13] Rahman G, Agarwal P, Mubeen S, ArshadM. Fractional integral operators involving extended
Mittag-Leer function as its kernel. Boletn de la Sociedad Matematica Mexicana 2018; 24:
381{392.
[14] Rainville ED. Special functions, New York: Chelsea Publ. Co.; 1960.
[15] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: Theory and Applications,
New York:Gordon and Breach, 1993.
[16] Saxena RK, Kalla SL, Saxena Ravi. Multivariate analogue of generalized Mittag-Leer function.
Integral Transforms and Special Functions 2011; 22(7): 533{548.
[17] Srivastava HM, Manocha HL. A treatise on generating functions, Halsted Press (Ellis Horwood
Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and
Toronto,1984
[18] Srivastava HM, Daoust MC. A note on the convergence of Kampe de feriet's double hypergeometric
series. Mathematische Nachrichten 1972; 53: 151{159.
[19] Srivastava HM, Daoust MC. On Eulerian integrals associated with Kampe de feriet's function.
Institut Matheematique(Belgrade) Nouvelle Seerie 1969; 23: 199{202.
[20] Srivastava HM,Tomovski Z. Fractional calculus with an integral operator containing a generalized
Mittag-Leer function in the kernel. Applied Mathematics and Computation 2009;
211: 198{210.
[21] Titchmarsh EC. Introduction to the Theory of Fourier Integrals, 3rd eds., New York: Chelsea;
1986.
[22] Wiman A. Uber den fundamental Satz in der Theorie der Funktionen E (x). Acta Mathematica
1905; 29: 191-201.
[2] Bin-Saad Maged Gumman, Anvar Hasanov, Michael Ruzhansky. Some properties related
to the Mittag{Leer function of two variables. Integral Transforms and Special Functions.
https://doi.org/10.1080/10652469.2021.1939328.
[3] Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Transcendental Functions, Vol.
I, New York, Toronto and London: McGraw-Hill Book Company, 1953.
[4] Goren
o R, Mainardi R. On Mittag-Leer function in fractional evaluation processes. Journal
of Computational and Applied Mathematics 2000; 118: 283-299.
[5] Goren
o R, Mainardi F, Srivastava HM. Special functions in fractional relaxation oscillation
and fractional di usion-wave phenomena, In Proceedings of the Eighth International Colloquium
on Di erential Equations (Plovdiv, Bulgaria; August 18{23, 1997), ed. D. Bainov,
195{ 202. Utrecht: VSP Publishers.; 1998.
[6] Kilbas AA, Saigo M. On Mittag-Leer type function, fractional calculus operators and solution
of integral equations. Integral Transforms and Special Functions 1996; 4: 355-370.
[7] Kilbas AA , Saigo M, Saxena RK. Generalized Mittag-Leer function and generalized fractional
calculus operators. Integral Transforms and Special Functions 2004; 15: 31-49.
[8] Kilbas AA, Saigo M. On Mittag-Leer type function, fractional calculus operators and solutions
of integral equations. Integral Transforms and Special Functions 1996; 4: 355{370.
[9] Kilbas AA, Saigo M, Saxena RK. Generalized Mittag-Leer function and generalized fractional
calculus operators. Integral Transforms and Special Functions 2004; 15: 31{49.
[10] Mittag-Leer GM. Sur la nouvelle fonction E (x). Comptes rendus de l'Academie des Sciences
1903; 137: 554-558.
[11] Podlubny I. Fractional di erential equations, San Diego:Academic Press, 1999.
[12] Prabhakar TR . A singular integral equation with a generalized Mittag-Leer function in the
kernel. Yokohama Mathematical Journal 1971; 19 :7-15.
[13] Rahman G, Agarwal P, Mubeen S, ArshadM. Fractional integral operators involving extended
Mittag-Leer function as its kernel. Boletn de la Sociedad Matematica Mexicana 2018; 24:
381{392.
[14] Rainville ED. Special functions, New York: Chelsea Publ. Co.; 1960.
[15] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: Theory and Applications,
New York:Gordon and Breach, 1993.
[16] Saxena RK, Kalla SL, Saxena Ravi. Multivariate analogue of generalized Mittag-Leer function.
Integral Transforms and Special Functions 2011; 22(7): 533{548.
[17] Srivastava HM, Manocha HL. A treatise on generating functions, Halsted Press (Ellis Horwood
Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and
Toronto,1984
[18] Srivastava HM, Daoust MC. A note on the convergence of Kampe de feriet's double hypergeometric
series. Mathematische Nachrichten 1972; 53: 151{159.
[19] Srivastava HM, Daoust MC. On Eulerian integrals associated with Kampe de feriet's function.
Institut Matheematique(Belgrade) Nouvelle Seerie 1969; 23: 199{202.
[20] Srivastava HM,Tomovski Z. Fractional calculus with an integral operator containing a generalized
Mittag-Leer function in the kernel. Applied Mathematics and Computation 2009;
211: 198{210.
[21] Titchmarsh EC. Introduction to the Theory of Fourier Integrals, 3rd eds., New York: Chelsea;
1986.
[22] Wiman A. Uber den fundamental Satz in der Theorie der Funktionen E (x). Acta Mathematica
1905; 29: 191-201.
Published
2025-02-04
Issue
Section
Articles
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
