Explicit representations of some generalized and mixed relatives of the Bessel functions

Special functions can be defined in different ways such as Rodrigue's formulae, generating functions, summation formulae, integral representations et cetera, but it is usually shown to be expressible as a series, because this is frequently the most practical way to obtain numerical values for the functions. In this paper, the explicit representations of certain mixed special functions related to the Bessel functions are obtained. A Laurent type hypergeometric generating relation is derived using series rearrangement technique. Some special cases are obtained as generating function of the known mixed type relatives of the Bessel functions. Finally explicit forms of these mixed type special functions are obtained as applications.