Generalized Mittag-Leffler-type function of arbitrary order and its properties related to integral transforms and fractional calculus
Abstract
This paper introduces a novel generalization of the Mittag-Leffler function, delving into its fundamental characteristics. The analysis encompasses a thorough exploration of its properties, including the derivation of recurrence relations, differential formulas, and various integral representations such as the Euler, Laplace, Mellin, Whittaker, and Mellin–Barnes transforms. Furthermore, the study establishes connections to other significant special functions, expressing the new generalization in terms of the Fox-Wright function, the generalized hypergeometric function, and the H-function. The paper also defines associated fractional integral and differential operators, highlighting the function’s relevance to fractional calculus. Several noteworthy special cases are derived from the main results, demonstrating the breadth and adaptability of this new function. This research provides a comprehensive framework for understanding the properties of this generalized Mittag-Leffler function and suggests its potential for applications in diverse areas, particularly within the realm of fractional analysis and its related fields.
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Ali, K. K., Abd El salam, M. A., Mohamed, E. M., A numerical technique for a general form ofnonlinear fractionalorder differential equations with the linear functional argument. IJNSNS. 22, 83–91, (2021).
Ali, K. K.; Abd El Salam, M. A.; Mohamed, E. M.; Samet, B.; Kumar, S.; Osman, M. S., Numerical solution for generalized nonlinearfractional integro-differential equations with linear functional arguments using Chebyshev series. Adv. Differ. Equ. 2020, 1-23, (2020).
Bin-Saad, M., Al-hashami, A., Younis, J., Some fractional calculus properties of bivariate Mittag-Leffler function. J. Frac. Calc. Appl. 14, 214–227, (2023).
Bin-Saad, M., Anvar, H., Ruzhansky, M., Some properties related to the Mittag-Leffler function of two variables. Integral Transforms Spec. Funct. 33, 400–418, (2022).
Khan, N., Khan, S., Xin, Q.; Tchier, F., Malik, S. N., Javed, U., Some applications of analytic functions associated with q-fractional operator. Mathematics. 11, 1–17, (2023).
Kilbas, A. A., Saigo, M., H-Transforms: Theory and Applications. London, New York: Chapman and Hall/CRC, (2004).
Kilbas, A.A., Sebastian, N., Generalized fractional integration of Bessel function of the first kind. Integral Transform. Spec. Funct. 19, 869–883, (2008).
Kiryakova, V., A guide to special functions in fractional calculus. Mathematics. 9, 1–40, (2021).
Mahdy, A. M., Stability, existence, and uniqueness for solving fractional glioblastoma multiforme using a Caputo-Fabrizio derivative. Math. Methods Appl. Sci. 48, 7360–7377, (2023).
Mahdy, A. M., Numerical solution and optimal control for fractional tumor immune model. J. Appl. Anal. Comput. 14, 3033–3045, (2024).
Mahdy, A. M.; Higazy, M.; Mohamed, M. S., Optimal and memristor-based control of a nonlinear fractional tumorimmune model Materials & Continua. 67, 3463–3486, (2021).
Mittag-Leffler, M. G., Sur la nouvelle fonction Eα(x). C. R. Acad. Sci. 137, 554–558, (1903).
Mohamed, D. S.; Abdou, M. A.; Mahdy, A. M., Dynamical investigation and numerical modeling of a fractional mixed nonlinear partial integro-differential problem in time and space. J. Appl. Anal. Comput. 14, 3458–3479, (2024).
Paneva-Konovska, J., Kiryakova, V., On the multi-index Mittag-Leffler functions and their Mellin transforms. Intern. J. Appl. Math. 33, 549–571, (2020).
Pathan, M. A., Bin-Saad, M., Mittag-leffler-type function of arbitrary order and their application in the fractional kinetic equation. Partial Differ. Equ. Appl. 4, 1–25, (2023).
Prabhakar, T. R., A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15, (1970).
Prajapati, J. C., Shukla, A. K., A remarkable survey on generalized Mittag-Leffler function and applications. Int. J. Math. Eng. And Sci. 1, 8–56, (2012).
Rainville, E. D., Special Functions. The Macmillan Company: New York, NY, USA, (1960).
Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, (1993).
Shahwan, M., Bin-Saad, M., Al-Hashami, A., Some properties of bivariate Mittag-Leffer function. J. Anal. 31, 2063–2083, (2023).
Shukla, A. K., Prajapati, J. C., On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336, 797–811, (2007).
Sneddon, I. N., The Use of Integral Transforms. Tata McGraw Hill, New Delhi, (1979).
Srivastava, H. M., Choi, J., Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers: Amsterdam, The Netherlands; London, UK; New York, NY, USA, (2012).
Srivastava, H. M., Manocha, H .L., A Treatise on Generating Functions. Halsted Press (Ellis Horwood Limited): Chichester, UK; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, ON, Canada, (1984).
Srivastava, H. M., Fernandez, A., Baleanu, D., Some new fractional-calculus connections between Mittag-Leffler functions. Mathematics. 7, 1-10, (2019).
Whittaker, E. T., Watson, G. N., A Course of Modern Analysis. Cambridge Univ. Press, Cambridge, (1962).
Wiman, A., ¨ Uber den fundamentalsatz in der theorie der funktionen Eα(x). Acta Math. 29, 191–201, (1905).
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