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- Title
Bispectrality of Multivariable Racah–Wilson Polynomials.
- Authors
Geronimo, Jeffrey; Iliev, Plamen
- Abstract
We construct a commutative algebra ${\mathcal{A}}_{x}$ 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, ). It is shown that for specific values of the variables x=( x1, x2,..., 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=( n1, n2,..., 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, ). 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, ), this change of variables and parameters in ${\mathcal{A}}_{x}$ and ${\mathcal{A}}_{n}$ leads to bispectral commutative algebras for the multivariable Wilson polynomials.
- Subjects
COMMUTATIVE algebra; RACAH algebra; POLYNOMIALS; DIFFERENCE operators; MATHEMATICAL analysis
- Publication
Constructive Approximation, 2010, Vol 31, Issue 3, p417
- ISSN
0176-4276
- Publication type
Article
- DOI
10.1007/s00365-009-9045-3