Abstract
We construct a commutative algebra \({\mathcal{A}}_{x}\) of difference operators in ℝp, 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 \({\mathcal{A}}_{n}\) 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 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 \({\mathcal{A}}_{x}\) and \({\mathcal{A}}_{n}\) leads to bispectral commutative algebras for the multivariable Wilson polynomials.
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Communicated by Edward B. Saff.
The first author was partially supported by an NSF grant.
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Geronimo, J.S., Iliev, P. Bispectrality of Multivariable Racah–Wilson Polynomials. Constr Approx 31, 417–457 (2010). https://doi.org/10.1007/s00365-009-9045-3
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DOI: https://doi.org/10.1007/s00365-009-9045-3
Keywords
- Bispectral problem
- Classical multivariable orthogonal polynomials
- Hypergeometric functions
- Askey-scheme