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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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