Mittag-Leffler based Bessel and Tricomi functions via umbral approach

Abstract

Various types of functions, their generalizations and extentions have been widely explored, especially for their applications in various fields of research. In this article, we show that the use of methods of an operational nature, such as umbral calculus, allows us to acquire the hybrid form of the Bessel and Tricomi functions in terms of Mittag-Leffler functions. Certain novel identities such as generating functions, series representations, differential equations, derivative formulae, summation formula and Jacobi-Anger expansion for Mittag-Leffler-Bessel functions are obtained. Some integral formulae for the Oth-order Mittag-Leffler-Bessel functions are established. In addition, the Mittag-Leffler-Tricomi functions are constructed and some captivating properties of these polynomials are explored. The natural transforms of the Mittag-Leffler-Bessel functions and Mittag-Leffler-Tricomi functions are investigated and Laplace and Sumudu transforms are obtained as special cases. The graphical representations of these hybrid functions are given for special values of the parameters.