Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

*(English)*Zbl 0977.33013It is well-known that irreducible representations of Lie groups and Lie algebras and of quantum groups are used for derivations of properties of special functions and orthogonal polynomials. Many properties of these functions and polynomials were derived on this base. The authors of the paper use the discrete series representations of the Lie algebra \(\text{su}(1,1)\) and of the quantized universal enveloping algebra \(U_q(\text{su}(1,1))\) for derivation of convolutions for orthogonal polynomials. The interpretation of the Meixner-Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra \(\text{su}(1,1)\) and the Clebsch-Gordan decomposition for this algebra 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 \(U_q(\text{su}(1,1))\), \(q\)-Racah polynomials are interpreted as Clebsch-Gordan coefficients for this algebra, and the linearization coefficients for a two-parameter family of Askey-Wilson polynomials are derived.

Reviewer: A.Klimyk (Kyïv)

##### MSC:

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |

33D80 | Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |

17B20 | Simple, semisimple, reductive (super)algebras |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |