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Orthogonality of \(q\)-polynomials for non-standard parameters. (English) Zbl 1229.33016

The paper presents a general tool, which provides a non-standard orthogonality property for any sequence of orthogonal polynomials \(\{p_n\}\) with some vanishing coefficient \(\gamma_n\) in the three-term recurrence relation:
\[ xp_n= p_{n+1} + \beta_n p_n + \gamma_n p_{n-1},\quad n \geq 0. \]
The tool is a degenerate version of the Favard theorem, and it is applied to obtain the orthogonality of \(q\)-polynomials for non-standard parameters. Indeed, orthogonality for the Askey-Wilson polynomials \(p_n(x;a,b,c,d;q)\) is known only when the product of any two parameters \(a,b,c,d\) is not a negative integer power of \(q\), and the orthogonality of the big \(q\)-Jacobi polynomials \(p_n(x;a,b,c;q)\) is known when not any of the numbers \(a,b,c,abc^{-1}\) is a negative integer power of \(q\). With the help of the technique mentioned before, the orthogonality properties of the Askey-Wilson and big \(q\)-Jacobi polynomials for almost all values of the parameters are obtained.
Finally, the results are also applied to derive non-standard orthogonality properties for continuous dual \(q\)-Hahn, big \(q\)-Laguerre, \(q\)-Meixner and little \(q\)-Jacobi polynomials for non-classical parameters.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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