Mehler Integral Transforms Associated with Jacobi Functions with Respect to the Dual Variable

We prove a Mehler representation for Jacobi functions w a , b . t. with respect to l the dual variable l. We exploit this representation to define a pair of dual integral transforms x and its transposed tx . We define two second order difference a,b a, b operators P and Q such that w a , b . t. is an eigenfunction of P with respect a,b l a , b to the dual variable l, and x and tx are permutation operators between P a,b a,b a, b and Q. Next we give some spaces of functions on which x and tx are a,b a, b isomorphisms and we establish inversion formulas for these transforms