Bispectrality of Multivariable Racah-wilson Polynomials

We construct a commutative algebra Ax of difference operators in R, depending on p + 3 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 Grünbaum [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.