Certain integrals involving generalized Mittag-Leffler type functions

DOI: https://doi.org/10.5937/vojtehg70-40296
Keywords: Mittag-Leffler function, Generalized hypergeometric function, Bessel Maitland function, Jacobi polynomials, Hermite polynomials

Abstract


Introduction/purpose: Certain integrals involving the generalized Mittag-Leffler function with different types of polynomials are established.

Methods: The properties of the generalized Mittag-Leffler function are used in conjunction with different kinds of polynomials such as Jacobi, Legendre, and Hermite in order to evaluate their integrals.

Results: Some integral formulae involving the Legendre function, the Bessel Maitland function and the generalized hypergeometric functions are derived. 

Conclusions: The results obtained here are general in nature and could be useful to establish further integral formulae involving other kinds of polynomials. 

References

Dorrego, G.A. & Cerutti R.A. 2012. The k-Mittag-Leffler function. International Journal of Contemporary Mathematics Sciences, 7(15), pp.705-716 [online]. Available at: http://www.m-hikari.com/ijcms/ijcms-2012/13-16-2012/ceruttiIJCMS13-16-2012-2.pdf [Accessed: 20 August 2022].

Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F.G. 1953a. Higher transcendental functions Volume 1 (Bateman Manuscript Project) [online]. New York, Toronto and London: McGraw-Hill Book Company. Available at: https://authors.library.caltech.edu/43491/19/Volume%201.pdf [Accessed: 20 August 2022].

Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F.G. 1953b. Higher transcendental functions Volume 3 (Bateman Manuscript Project) [online]. New York, Toronto and London: McGraw-Hill Book Company. Available at: https://authors.library.caltech.edu/43491/10/Volume%203.pdf [Accessed: 20 August 2022].

Faraj, A.W., Salim, T.O., Sadek, S. & Ismail, J. 2013. Generalized Mittag-effler Function Associated with Weyl Fractional Calculus Operators. Journal of Mathematics, 2013(art ID:821762). Available at: https://doi.org/10.1155/2013/821762

Gehlot, K.S. 2021. The generalized k-Mittag-Leffler function. International Journal of Contemporary Mathematical Sciences, 7, pp.2213-2219.

Haq, S., Khan, A.H. & Nisar, K.S. 2019. A study of new class of integrals associated with generalized Struve function and polynomials. Communications of the Korean Mathematical Society, 34(1), pp.169-183. Available at: https://doi.org/10.4134/CKMS.c170490

Khan, M.A. & Ahmed, S. 2012. Fractional calculus operators involving generalized Mittag-Leffler function. World Applied Programming, 2(12), pp.492-499.

McBride, A.C. 1995. V. Kiryakova Generalized fractional calculus and applications (Pitman Research Notes in Mathematics Vol. 301, Longman1994), 388 pp., 0 582 21977 9, £39. Proceedings of the Edinburgh Mathematical Society, 38(1), pp.189-190. Available at: https://doi.org/10.1017/S0013091500006325

Mittag-Leffler, G.M. 1903. Sur la nouvelle fonction Eα(x). CR Acad. Sci. Paris, 137(2), pp.554-558.

Nadir, A., Khan, A. & Kalim, M. 2014. Integral transforms of the generalized Mittag-Leffler function. Applied Mathematical Science, 8(103), pp.5145-5154. Available at: https://doi.org/10.12988/ams.2014.43218

Prabhakar, T.R. 1971. A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Mathematical Journal, 19(1), pp.7-15 [online]. Available at: https://ynu.repo.nii.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=6514&item_no=1&page_id=15&block_id=22 [Accessed: 20 August 2022].

Prajapati, J.C., Jana, R.K., Saxena, R.K. & Shukla, A.K. 2013. Some results on the generalized Mittag-Leffler function operator. Journal of Inequalities and Applications, 2013(art.number:33). Available at: https://doi.org/10.1186/1029-242X-2013-33

Prajapati, J.C. & Shukla, A.K., 2012. Decomposition of Generalized Mittag-Leffler Function and Its Properties. Advances in Pure Mathematics, 2(1), p.8-14. Available at: https://doi.org//10.4236/apm.2012.21003

Purohit, S.D., Kalla, S.L. & Suthar, D.L. 2011. Fractional integral operators and the multiindex Mittag-Leffler functions. SCIENTIA Series A: Mathematical Sciences, 21, pp.87–96 [online]. Available at: http://scientia.mat.utfsm.cl/archivos/vol21/vol21art9.pdf [Accessed: 20 August 2022].

Rainville, E.D. 1960. Special functions (Vol. 5). New York: The Macmillan Company.

Salim, T.O. & Faraj, A.W. 2012. A generalization of Mittag-Leffler function and integral operator associated with fractional calculus. Journal of Fractional Calculus and Applications, 3(5), pp.1-13 [online]. Available at: https://www.naturalspublishing.com/download.asp?ArtcID=1893 [Accessed: 20 August 2022].

Saxena, V.P. 2008. The I-function. New Delhi: Anamaya publisher.

Saxena, R.K., Pogany, T.K., Ram, J. & Daiya, J. 2011. Dirichlet Averages of Generalized Multi-index Mittag-Leffler Functions. Armenian Journal of Mathematics, 3(4), pp.174-187 [online]. Available at: http://armjmath.sci.am/index.php/ajm/article/view/79 [Accessed: 20 August 2022].

Shukla A.K. & Prajapati J.C. 2007. On a generalization of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Applications, 336(2), pp.797-811. Available at: https://doi.org/10.1016/j.jmaa.2007.03.018

Singh, D.K. & Rawat, R.A.H.U.L. 2013. Integrals involving generalized Mittag-Leffler function. Journal of Fractional Calculus and Applications, 4(2), pp.234-244 [online]. Available at: http://math-frac.org/Journals/JFCA/Vol4(2)_July_2013/Vol4(2)_Papers/07_Vol.%204(2)%20July%202013,%20No.%207,%20pp.%20234-244..pdf [Accessed: 20 August 2022].

Srivastava, H.M. & Manocha, H.L. 1984. A Treatise on Generating Functions. New York: Hasted Press.

Srivastava, H.M. & Tomovski, Ž. 2009. Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel. Applied Mathematics and Computation, 211(1), pp.198-210. Available at: https://doi.org/10.1016/j.amc.2009.01.055

Wiman, A., 1905. Über den Fundamentalsatz in der Teorie der Funktionen Eυ(x). Acta Mathematica, 29, pp.191-201. Available at: https://doi.org/10.1007/BF02403202

Wright, E.M. 1935a. The Asymptotic Expansion of the Generalized Bessel Function. Proceedings of the London Mathematical Society, s2-38(1), pp.257-270. Available at: https://doi.org/10.1112/plms/s2-38.1.257

Wright, E.M. 1935b. The Asymptotic Expansion of the Generalized Hypergeometric Function. Journal of the London Mathematical Society, s1-10(4), pp.286-293. Available at: https://doi.org/10.1112/jlms/s1-10.40.286

Published
2022/10/14
Issue
Vol. 70 No. 4 (2022): October - December
Section
Original Scientific Papers

Copyright (c) 2022 Sirazul Haq, Maggie Aphane, Mohammad Saeed Khan, Nikola Fabiano

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