S-Generalized Mittag-Leffler Function and its Certain Properties

Abstract

In 2014, a new generalized beta function which consist of seven parameters, defined and studied bySrivastava et al. [H. M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized Gausshypergeometric functions, Appl. Math. Comput., 247 (2014), pp. 348-352]. In 2015, Srivastava et al. [H.M. Srivastava, R. Jain and M. K. Bansal, A study of the S-generalized Gauss hypergeometric functionand its associated integral transforms, Turkish J. Anal. Number Theory, 3 (2015), pp. 101-104] called thisgeneralization as ”S-generalized beta function“ and use it to define S-generalized Gauss hypergeometricfunction. In this paper, by using S-generalized beta function, we introduce a new generalization ofMittag-Leffler function. This new generalization of Mittag-Leffler function is consist of eleven parameters.We also investigate some of its certain properties such as integral representations, recurrence formulasand derivative formulas by using classical and fractional derivatives. Furthermore, we determine itsMellin, beta and Laplace integral transforms.