We establish the generalized fractional derivative operators of the product of two generalized multi-index Mittag-Leffler functions. Further, the Riemann-Liouville, Kober and Saigo fractional derivative operators of given functions are obtained.

generalized fractional derivative operators, multi-index Mittag-Leffler function.

Received: October 22, 2023; Accepted: December 11, 2023; Published: December 18, 2023

How to cite this article: Sunil Kumar and Krishna Gopal Bhadana, Generalized fractional derivative operators of the product of two multi-index Mittag-Leffler functions with applications, Far East Journal of Applied Mathematics 116(4) (2023), 407-419. http://dx.doi.org/10.17654/0972096023020

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] A. Erdelye et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, London, 1953.
[2] A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of first kind, Integral Transform Spec. Funct. 19 (2008), 869-883.
[3] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336(2) (2007), 797-811.
[4] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function II, Proc. London Math. Soc. 46(2) (1940), 389-408.
[5] F. Oberhettinger, Tables of Mellin Transforms, Springer, New York, 1974.
[6] H. M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6.
[7] J. Choi, D. Kumar and S. D. Purohit, Integral formulas involving a product of generalized Bessel functions of the first kind, Kyungpook Math. J. 56(1) (2016), 131-136.
[8] J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, Filomat 30(7) (2016), 1931-1939.
[9] J. L. Lavoie and G. Trottier, On the sum of certain Appell’s series, Ganita 20 (1969), 43-46.
[10] K. K. Kataria and P. Vellaisamy, The k-Wright function and Marichev-Saigo-Maeda fractional operators, J. Anal. 23 (2015), 75-87.
[11] M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1978), 135-143.
[12] M. Saigo and N. Maeda, More generalization of fractional calculus, Transform Method & Special Functions, Bulgarian Academy of Sciences, Sofia, Vol. 96, 1998, pp. 386-400.
[13] R. K. Saxena and R. K. Parmar, Fractional integration and differentiation of the generalized Mathieu series, Axioms 6 (2017), 18.
[14] R. K. Saxena and K. Nishimoto, N-fractional calculus of generalized Mittag- Leffler functions, J. Fract. Calc. 37 (2010), 43-52.
[15] R. K. Saxena and M. Saigo, Generalized fractional calculus of the H-function associated with the Appell function J. Fract. Calc. 19 (2001), 89-104.
[16] T. Pohlen, The Hadamard product and universal power series, Ph.D. Thesis, Universität Trier, Trier, Germany, 2009.
[17] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15.
[18] V. N. Mishra, D. L. Suthar and S. D. Purohit, Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function, Cogent Mathematics 4 (2017), 1320830.
https://doi.org/10.1080/23311835.2017.1320830.