On fractional calculus of the bivariate Mittag-Leffler function
On fractional calculus of the bivariate Mittag-Leffler function
Resumo
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.
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Referências
[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.
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2025-02-04
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