Bispectrality of multivariable Racah-Wilson polynomials
Abstract
We construct a commutative algebra Ax of difference operators in Rp, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials Rp(n;x) considered by Tratnik [27]. It is shown that for specific values of the variables x=(x1,x2,...,xp) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra An in the variables n=(n1,n2,...,np) which is also diagonalized by Rp(n;x). Thus Rp(n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials [26], this change of variables and parameters in Ax and An leads to bispectral commutative algebras for the multivariable Wilson polynomials.