MODELS OF Q-ALGEBRA REPRESENTATIONS - Q-INTEGRAL TRANSFORMS AND ADDITION THEOREMS

In his classic book on group representations and special functions Vil enkin studied the matrix elements of irreducible representations of th e Euclidean and oscillator Lie algebras with respect to countable base s of eigenfunctions of the Cartan subalgebras, and he computed the sum mation identities for Bessel functions and Laguerre polynomials associ ated with the addition theorems for these matrix elements. He also stu died matrix elements of the pseudo-Euclidean and pseudo-oscillator alg ebras with respect to the continuum bases of generalized eigenfunction s of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as inte gral transform identities for Bessel functions and for confluent hyper geometric functions. Here we work out q analogs of these results in wh ich the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings E(q) and e(q). Th is study of representations of the Euclidean quantum algebra and the q -oscillator algebra (not a quantum algebra) leads to summation, integr al transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.