Harmonic analysis associated with the Jacobi-Dunkl operator on ]-π/2, π/2[

We consider a differential-difference operator Λα,β, α ≥ β ≥ - 1/2; α ≠ -1/2; on] - π/2, π/2[. 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.