Rogers's \(q\)-ultraspherical polynomials on a quantum \(2\)-sphere (Q1814228)

This article gives a realization of the continuous \(q\)-ultraspherical polynomials \(C^ \lambda_ n(x;q)\) as the spherical functions on a quantum 2-sphere \(S=S^ 2_ q(1,1)\) of Podle\`s. This result is obtained by combining the work of Koornwinder on realiztion of continuous \(q\)-Legendre polynomials \(C_ n^{1/2}(x;q)\) as spherical matrix elements on the quantum group \(SU_ q(2)\) and the work of the authors on realization of the symmetric big \(q\)-Jacobi polynomials on the quantum 2- sphere of Podle\`s. The algebra \(A(S)\) of functions on \(S\) is decomposed into the direct sum of irreducible \(A(G)\)-comodules \(V_ j\) of spin \(j\in\mathbb{N}\). Each \(V_ j\) is decomposed by eigenvectors \(\psi^ j_ m\) of the operator \(D\) of Koornwinder. The latter are expressed by continuous \(q\)-ultraspherical polynomials \(C^ \lambda_ n(x;q)\) with \(\rho={\sqrt -1}(\xi+q\eta)/2\). The following three-term recurrence relation is used: \[ 2\rho(1-q^{4j+2})\psi^ j_ m=(1-q^{4(j- m+1)})\psi_ m^{j+1}+(1-q^{4(j+m)})\psi_ m^{j-1}. \]