Harmonic analysis associated with the Jacobi-Dunkl operator on ]-π2,π2[

We consider a differential-difference operator Λα,β, α⩾β⩾-12, α≠-12 on ]-π2,π2[. The eigenfunction of this operator equal to 1 at zero is related to the Jacobi polynomials and to their derivatives. We give a Laplace integral representation for this function called the Jacobi-Dunkl polynomial. Next we study the harmonic analysis associated with the operator Λα,β.