Quantum Algebra ε(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 eq(2). The quantum algebra eq(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.