Non-symmetric Jacobi and Wilson type polynomials

Abstract

Consider a root system of type $B{C}_1$ on the real line $\mathbb{R}$ with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an $L^2$-space on $\mathbb{R}$ to a $L^2$-space of ${\mathbb{C}}^2$-valued functions on ${\mathbb{R}}^{+}$ with the Harish-Chandra measure $|c(\lam)|^{-2}d\lam$. By introducing a weight function of the form $\cosh^{-\sig}(t)\tanh^{2k} t$ on $\mathbb{R}$ we find an orthogonal basis for the $L^2$-space on $\mathbb{R}$ consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed $\sig$ and $k$). We find a Rodrigues type formula for the functions in terms of the Cherednik operator. We compute explicitly their Cherednik-Opdam transforms. We discover thus a new family of ${\mathbb{C}}^2$-valued orthogonal polynomials. In the special case when $k=0$ the even polynomials become Wilson polynomials, and the corresponding result was proved earlier by Koornwinder.

References in corpus (2)

• Properties of some families of hypergeometric orthogonal polynomials in several variables
• Fourier transforms related to a root system of rank 1