Bispectrality of Multivariable Racah–Wilson Polynomials

We construct a commutative algebra of difference operators in ℝ p , depending on p +3 parameters, which is diagonalized by the multivariable Racah polynomials R p ( n ; x ) considered by Tratnik (J. Math. Phys. 32(9):2337–2342, 1991 ). It is shown that for specific values of the variables x =( x 1 , x 2 ,…, x p ) there is a hidden duality between n and x . Analytic continuation allows us to construct another commutative algebra in the variables n =( n 1 , n 2 ,…, n p ) which is also diagonalized by R p ( n ; x ). Thus, R p ( n ; x ) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grünbaum (Commun. Math. Phys. 103(2):177–240, 1986 ). Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials (Tratnik in J. Math. Phys. 32(8):2065–2073, 1991 ), this change of variables and parameters in and leads to bispectral commutative algebras for the multivariable Wilson polynomials.