Bispectrality of Multivariable Racah–Wilson Polynomials

By Jeffrey S. Geronimo, Plamen Iliev. 2007

73 citations recorded by the source corpus.

AbstractWe 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 Rp(n;x) considered by Tratnik (J. Math. Phys. 32(9):2337–2342, 1991). 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 ${\mathcal{A}}_{n}$ 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 (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.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0112295aea095d527243180184252a9bb4854872