Non-symmetric Jacobi and Wilson Type Polynomials

Consider a root system of type BC1 on the real line R with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an L-space on R to a L-space of C-valued functions on R with the Harish-Chandra measure |c(λ)|dλ. By introducing a weight function of the form cosh(t) tanh t on R we find an orthogonal basis for the L-space on R consisting of even and odd functions expressed in terms of the Jacobi polynomials (for each fixed σ 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 C-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.