POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$

We consider a differential-difference operator Λα,β, $\alpha > -\frac{1}{2}$, $\beta\in {\mathbb R}$ on ${\mathbb R}$. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for $\alpha\ge\beta\ge -\frac{1}{2}$, $\alpha\ne -\frac{1}{2}$, the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).