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The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here, we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag–Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation. [ABSTRACT FROM AUTHOR]

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed. [ABSTRACT FROM AUTHOR]