Five-parameter discrete Mittag-Leffler function and corresponding discrete double fractional operator

Abstract

In recent years, discrete Mittag-Leffler (ML) functions have been investigated intensively since they are connected with the discrete fractional calculus and have interesting applications in applied sciences. In this study, inspired by the five-parameter ML function by Özarslan et al. [16], we define its discrete version, namely five-parameter discrete ML function. We explore various characteristics, such as the double fractional sum and difference, of the discrete ML function with five parameters. Then, using the mentioned discrete function, we introduce the discrete double fractional operator having this function through the semigroup property. We also obtain the left inverse operator which is the corresponding discrete difference operator for the discrete double fractional operator.

Keywords

• discrete ML functions
• discrete fractional sums
• discrete fractional differences
• semigroup property

References

1. [1] T. Abdeljawad, Different type kernel h−fractional differences and their fractional h−sums, Chaos Solit. Fract. 116, 146-156, 2018.
2. [2] T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Solit. Fract. 126 (2), 315-324, 2019.
3. [3] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ. 2016, 2016.
4. [4] F. Atici and P. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ. 2 (2), 165-176, 2007.
5. [5] A. Fernandez, D. Baleanu and H.M. Srivastava, Series representations for models of fractional calculus involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simul. 67, 517-527, 2019.
6. [6] A. Fernandez, C. Kürt, and M.A. Özarslan, A naturally emerging bivariate Mittag- Leffler function and associated fractional-calculus operators, Comput. Appl. Math. 39 (3), 2020.
7. [7] C.S. Goodrich and A.C. Peterson, Discrete fractional calculus, Springer, Berlin, 2015.
8. [8] A.A. Kilbas, M. Saigo and J.J. Trujillo, On the generalized Wright function, Frac. Calc. Appl. Anal. 5 (4), 437-460, 2002.

9. [9] A.A. Kilbas, H.M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
10. [10] K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York, NY, USA, 1993.
11. [11] P.O. Mohammed, A generalized uncertain fractional forward difference equations of Riemann-Lioville type, J. Math. Res. 11 (4), 43-50, 2019.
12. [12] P.O. Mohammed, T. Abdeljawad and F.K. Hamasalh, On Riemann-Liouville and Caputo fractional forward difference monotonicity analysis, Mathematics 9 (11), 1303, 2021.
13. [13] P.O. Mohammed, T. Abdeljawad and F.K. Hamasalh, Discrete Prabhakar fractional difference and sum operators, Chaos Solit. Fract. 150, 111182, 2021.
14. [14] P.O. Mohammed, F.K. Hamasalh and T. Abdeljawad, Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels, Adv. Differ. Equ. 2021, 213, 2021.
15. [15] K.B. Oldham and J. Spanier, The Fractional Calculus; Academic Press, San Diego, 1974.
16. [16] M.A. Özarslan and A. Fernandez, On a five-parameter Mittag-Leffler function and the corresponding bivariate fractional operators, Fractal Fract. 5 (2), 45, 2021.
17. [17] M.A. Özarslan and C. Kürt, On a double integral equation including a set of two variables polynomials suggested by Laguerre polynomials, Journal of Computational and Applied Analysis, 22 (7), 1198-1207, 2017.
18. [18] M.A. Özarslan and C. Kürt, Bivariate Mittag-Leffler functions arising in the solutions of convolution integral equation with 2D-Laguerre-Konhauser polynomials in the kernel, Appl. Math. Comput. 347, 631-644, 2019.
19. [19] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19, 7-15, 1971.
20. [20] S.G. Samko, A.A. Kilbas and O.I Marichev, Fractional Integrals and derivatives: theory and applications, Taylor & Francis, London, 1987 [orig. ed. in Russian: Nauka i Tekhnika. Minsk, 1987].
21. [21] R.K. Saxena, S.L. Kalla and R. Saxena, Multivariate analogue of generalised Mittag- Leffler function, Integral Transforms and Special Functions, 22 (7), 533-548, 2011.