QUANTUM ALGEBRA εq (2) AND 2D q-BESSEL FUNCTIONS

The main objective of this article is to introduce the 2D q-Bessel functions by means of the generating function and to frame these q-special functions into the context of the representation of the 2-dimensional Euclidian quantum algebra epsilon(q)(2). The quantum algebra epsilon(q)(2) and its representation present an algebraic derivation for an explicit expression of the 2D q-Bessel functions in terms of the product of the q-Bessel functions of the first and second kinds. The series expansion, differential recurrence relations and functional identities for these functions are derived. Further, the q-Hermite-Bessel functions are introduced and their series expansions are derived. These expansions are used to derive the hypergeometric representation and generating equation for the q-Hermite-Bessel functions.