Convolutions for Orthogonal Polynomials from Lie and Quantum Algebra Representations

Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra ${\frak{su}}(1,1)$ and the Clebsch--Gordan decomposition 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 quantized universal enveloping algebra for ${\frak{su}}(1,1)$, convolution identities for the Al-Salam and Chihara polynomials and the Askey--Wilson polynomials are derived by using the Clebsch--Gordan and Racah coefficients. For the quantized universal enveloping algebra for ${\frak{su}}(2)$, q-Racah polynomials are interpreted as Clebsch--Gordan coefficients, and the linearization coefficients for a two-parameter family of Askey--Wilson polynomials are derived.