Bivariate Knop–Sahi and Macdonald polynomials related to q-ultraspherical functions

AbstractKnop and Sahi introduced a family of non-homogeneous and non-symmetric polynomials, Gα(x;q,t), indexed by compositions. An explicit formula for the bivariate Knop–Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macdonald polynomials, and the bivariate Knop–Sahi polynomials are explicitly determined using the theory of basic hypergeometric series