Harmonic analysis associated with the Jacobi–Dunkl operator on

We consider a differential-difference operator Λ α , 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 Λ α , β .