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.

各种类型的函数及其概括和扩展已被广泛探索，特别是它们在各个研究领域的应用。在本文中，我们展示了使用运算性质的方法（例如本影微积分）使我们能够根据 Mittag-Leffler 函数获得 Bessel 和 Tricomi 函数的混合形式。获得了Mittag-Leffler-Bessel函数的生成函数、级数表示、微分方程、导数公式、求和公式以及Jacobi-Anger展开等一些新颖的恒等式。建立了O阶Mittag-Leffler-Bessel函数的积分公式。此外，还构建了 Mittag-Leffler-Tricomi 函数，并探索了这些多项式的一些迷人特性。研究了Mittag-Leffler-Bessel函数和Mittag-Leffler-Tricomi函数的自然变换，并获得了Laplace和Sumudu变换作为特例。这些混合函数的图形表示是针对参数的特殊值给出的。