Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

Iliev, Plamen

doi:10.1090/S0002-9947-2010-05183-9

Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
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Trans. Amer. Math. Soc. 363 (2011), 1577-1598 Request permission

Abstract:

We construct a commutative algebra $\mathcal A_{z}$, generated by $d$ algebraically independent $q$-difference operators acting on variables $z_1,z_2,\dots ,z_d$, which is diagonalized by the multivariable Askey-Wilson polynomials $P_n(z)$ considered by Gasper and Rahman (2005). Iterating Sears’ ${}_4\phi _3$ transformation formula, we show that the polynomials $P_n(z)$ possess a certain duality between $z$ and $n$. Analytic continuation allows us to obtain another commutative algebra $\mathcal A_{n}$, generated by $d$ algebraically independent difference operators acting on the discrete variables $n_1,n_2,\dots ,n_d$, which is also diagonalized by $P_n(z)$. This leads to a multivariable $q$-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.

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