2005/11/29 by Lizhong Peng, Peng, Lizhong, Genkai Zhang +1
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.math/0511709
openalex publication_date 2005/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a root system of type BC1 on the real line \mathbb R with general positive multiplicities. The Cherednik-Opdam transform defines a unitary operator from an L2-space on \mathbb R to a L2-space of \mathbb C2-valued functions on \mathbb R+ with the Harish-Chandra measure |c(\lam)|-2d\lam. By introducing a weight function of the form \cosh-\sig(t)\tanh2k t on \mathbb R we find an orthogonal basis for the L2-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 C2-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.