Bispectrality of multivariable Racah-Wilson polynomials

Details Bispectrality of multivariable Racah-Wilson polynomials Bispectrality of multivariable Racah-Wilson polynomials

2007-05-10

arxiv.org/abs/0705.1469v3

Constr. Approx. 31 (2010), no. 3, 417--457.

Mathematics

Classical Analysis and ODEs

minor corrections

Scientific paper

We construct a commutative algebra A_x of difference operators in R^p, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials R_p(n;x) considered by Tratnik [27]. 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 A_n 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 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 A_x and A_n leads to bispectral commutative algebras for the multivariable Wilson polynomials.

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