Harmonic analysis associated with the Jacobi-Dunkl operator on \(]-\frac{\pi}{2},\frac{\pi}{2}[\) (Q1772332)

In this paper the author considers the Jacobi-Dunkl operator \(\Lambda_{\alpha,\beta}\) on \(]-{\pi\over 2},{\pi\over 2}[\) given by \[ \Lambda_{\alpha,beta} f(\theta)= {d\over d\theta} f(\theta)+ {A_{\alpha,\beta}'(x)\over A_{\alpha,\beta}(x)} \Biggl({f(\theta)- f(-\theta)\over 2}\Biggr), \] where \(A_{\alpha,\beta}(\theta)= 2^{2\rho}(\sin|\theta|)^{2\alpha+ 1}(\cos\theta)^{2\beta+1}\), \(\alpha\geq \beta>-{1\over 2}\), \(\alpha\neq -{1\over 2}\), \(\rho= \alpha+ \beta+ 1\). The eigenfunctions of this operator equal to 1 at zero are called Jacobi-Dunkl polynomials. He proves for these polynomials a Laplace integral representation, and with the aid of these polynomials he defines the Jacobi-Dunkl transform and he establishes for this transform an inversion formula and a Parseval formula and a Paley-Wiener theorem.