Quantum 2-spheres and big q-Jacobi polynomials (Q913050)

This article gives a quantum-group theoretic interpretation of the big q- Jacobi polynomials \(P_ n^{(\alpha,\alpha)}(x;c,d;q)\) of symmetric type. Namely, they appear as spherical functions on the quantum 2-spheres \(S^ 2_ q(c,d)\) of P. Podles. This is shown via the construction of explicit orthogonal bases for Podles' function algebra for \(S^ 2_ q\) in terms of big q-Jacobi polynomials, and by Rodrigues' formula for the latter. The q-Hahn polynomials of symmetric type are interpreted similarly.