Generalized Mittag-Leffler Function and Its Properties

Abstract

Recently, Srivastava, C¸ etinkaya and Kıymaz [18] defined the generalized Pochhammer symbol and obtained some relations.In this paper, we define the generalized Mittag-Leffler function via the generalized Pochammer symbol and present some recurrencerelation, derivative properties, integral representation. Moreover, we obtain a relation between wright hypergeometric function and thegeneralized Mittag-Leffler function

Keywords

• Pochhammer symbol, generalized Mittag-Leffler function, Wright Hypergeometric function

Kaynakça

1. Agarwal, R. P: A propos d’ une note de M. Pierre Humbert, Comptes Rendus de l’ Academie des Sciences, vol. 236, pp. 203-2032, 1953.
2. Chaudhry M.A, Srivastava H.M, Paris R.B : Extended hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation, 159 (2004) 589-602.
3. Chaudhry M.A, Zubair, S.M: On a Class of Incomplete Gamma Functions with Applications. Haubold H. J., Mathai A. M., and Saxena R. K: Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, Vol 2011, 51 pages.
4. Humbert P. and Agarwal, R. P : Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bulletin of Science and Mathematics Series II, vol. 77, pp.180-185, 1953.
5. Kurulay M, Bayram M: Some properties of the Mittag-Leffler functions and their relation with the Wright function, Advance Difference Equations 2012, 2012:178.
6. Mittag-Leffler, G. M: Une generalisation de l’ integrale de Laplace-Abel, Comptes Rendus de l’ Academie des Sciences Serie II, vol. 137, pp. 537-539, 1903.
7. Mittag-Leffler,G. M: Sur la nouvelle fonction Eα(x),Comptes Rendus de l’ Academie des Sciences, vol. 137, pp. 554-558, 1903.
8. Mittag-Leffler, G. M, Mittag-Leffler, Sur la representation analytiqie d’une fonction monogene (cinquieme note), Acta Mathematica, vol. 29, no. 1, pp. 101-181, 1905.

9. ¨Ozarslan M.A : Some Remarks on Extended Hypergeometric, Extended Cofluent Hypergeometric and Extended Appell’s Functions, Journal of Compuattional Analysis and Applications, Vol. 14, NO:6, 1148-1153, 2012.
10. ¨Ozarslan M.A, ¨Ozergin E: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Mathematical and ComputerModelling, 52 (2010) 1825-1833.
11. Prabhakar T. R. : A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), pp. 7-15.
12. Samko, S. G, Kilbas,A. A. and Marichev, O. I: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
13. Srivastava H.M, Manocha, H. L: A Treatise on Generating Functions. Wiman, A :Uber den fundamentalsatz in der theorie der funktionen Eα(x), Acta Math., Vol. 29, p.p. 191-201, 1905.
14. Wiman, A: ¨Uber die Nullstellun der Funktionen Eα(x),Acta Mathematica, vol. 29, pp. 217-234, 1905.
15. ¨Ozarslan, M. A, Yılmaz Yas¸ar, B: The Extended Mittag-Leffler’s Function and Its Properties, Journal of Inequalities and Applications, 2013.
16. Srivastava, H.M, C¸ etinkaya, A, Kıymaz, O: A certain generalized Pochammer symbol and ıts applications to hypergeometric functions, Applied Mathematics and Computation, 226 (2014) 484-491.