The structure relation for Askey-Wilson polynomials
This paper gives in an elegant and easy way, a structure relation for Askey-Wilson polynomials in the form \( Lp_n = a_n p_{n+1} = c_n P_{n-1},\) where \(L\) is a linear operator which is skew symmetric with respect to the inner product \( \langle .,. \rangle\) and for which the Askey-Wilson polynomials are orthogonal,\(\langle Lf,g \rangle = - \langle f, Lg \rangle \) and \[ \begin{multlined} (Lf)[z] := ((1 - az)(1-bz)(1-cz)(1-dz)z^{-2}f[qz] - \\ (1 - a/z)(1 - b/z)(1-c/z)(1-d/z)z^{-2}f[q^{-1}z] (z-z^{-1})^{-1}. \end{multlined} \] It sends symmetric Laurent polynomials of degree \(n\) to symmetric Laurent polynomials of degree \(n+1\). A structure relation of this form can be obtained for families of orthogonal polynomials in the \(q\)-Askey scheme and the Askey scheme. The coefficients in the structure relation are very close to the coefficients in the three-term recurrence relation. The author discusses the relationship with bispectral problems, Zhedanov's algebra, and the string equation in the context of discrete matrix methods. Further specializations to continuous \(q\)- Jacobi, \(q\)-ultraspherical, and big \(q\)-Jacobi polynomials [see \textit{J. C. Medem, R. Álvarez-Nodarse} and \textit{F. Marcellán}, J. Comp. Appl. Math. 135, 157--196 (2001; Zbl 0991.33007)] are presented.
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