This proposal deals with the interplay between quantum groups and special functions. Special functions have their origin in the solutions of partial differential equations of mathematical physics in the 18th and 19th century and were introduced by Euler, Gauss, Riemann, Kummer, Bessel, etc. Since the work of Weyl, Bargmann, Gelfand, Wigner, etc., the interplay between special functions and group theory has become fruitful in both directions. In particular, the study of spherical functions on Lie groups has been related to special functions. Spherical functions on a Lie group G are functions which are left and right invariant with respect to a compact subgroup K. This interpretation leads to various properties such as integral representations, product formulas, eigenfunctions for differential operators, expansion formulas, etc. For specific examples the spherical functions can be matched explicitly to special functions and orthogonal polynomials. In turn this has been extended recently to the study of matrix-valued spherical functions where higher-dimensional representations of K are involved. For specific examples of groups the matrix-valued spherical functions have been identified with matrix-valued orthogonal polynomials, which have been studied from an analytic point of view since the 50s. Quantum groups are deformations of groups and there is a close relation to scalar-valued special functions of basic hypergeometric type. The goal is to introduce and study matrix-valued spherical functions on quantum groups and explicit examples leading to matrix-valued special functions of basic hypergeometric type, a new class of special functions. |