CONVOLUTIONS FOR ORTHOGONAL POLYNOMIALS FROM LIE AND QUANTUM ALGEBRA REPRESENTATIONS

The interpretation of the Meixner-Pollaczek, Meixner, and Laguerre pol ynomials as overlap coefficients in the positive discrete series repre sentations of the Lie algebra su(1, 1) and the Clebsch-Gordan decompos ition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the q uantized universal enveloping algebra for su(1; 1), convolution identi ties for the Al-Salam and Chihara polynomials and the Askey-Wilson pol ynomials are derived by using the Clebsch-Gordan and Racah coefficient s. For the quantized universal enveloping algebra for su(2), q-Racah p olynomials are interpreted as Clebsch-Gordan coefficients, and the lin earization coefficients for a two-parameter family of Askey-Wilson pol ynomials are derived.