MEHLER INTEGRAL-TRANSFORMS ASSOCIATED WITH JACOBI FUNCTIONS WITH RESPECT TO THE DUAL VARIABLE

We prove a Mehler representation for Jacobi functions phi(lambda)((alp ha,beta))(t) with respect to the dual variable lambda. We exploit this representation to define a pair of dual integral transforms chi(alpha ,beta) and its transposed (t) chi(alpha,beta). We define two second or der difference operators P-alpha,P-beta and Q such that phi(lambda)((a lpha,beta))(t) is an eigenfunction of P-alpha,P-beta with respect to t he dual variable lambda, and chi(alpha,beta) and (t) chi(alpha,beta) a re permutation operations between P-alpha,P-beta and Q. Next we give s ome spaces of functions on which chi(alpha,beta) and (t) chi(alpha,bet a) are isomorphisms and we establish inversion formulas for these tran sforms. (C) 1997 Academic Press.