Orthogonality of \(q\)-polynomials for non-standard parameters

Abstract

\(q\)-polynomials can be defined for all the possible parameters, but their orthogonality properties are unknown for several configurations of the 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 \(q\). Also, the orthogonality of the big \(q\)-Jacobi, \(p_n(x;a,b,c|q)\), is known when \(a, b, c, abc^{-1}\) is not a negative integer power of \(q\). In this paper, we obtain orthogonality properties for the Askey–Wilson polynomials and the big \(q\)-Jacobi polynomials for the rest of the parameters and for all \(n\in \mathbb N_0\). For a few values of such parameters, the three-term recurrence relation \[ xp_n=p_{n+1}+\beta_n\, p_n+\gamma_n \,p_{n-1},\quad n\ge 0, \] presents some index for which the coefficient \(\gamma_n=0\), and hence Favard’s theorem cannot be applied. For this purpose, we state a degenerate version of Favard’s theorem, which is valid for all sequences of polynomials satisfying a TTRR even when some coefficient \(\gamma_n\) vanishes, i.e., \(\{n\ :\gamma_n=0\}\ne \emptyset\).

We also apply this result to the continuous dual \(q\)-Hahn, big \(q\)-Laguerre, \(q\)-Meixner, and little \(q\)-Jacobi polynomials, although it is also applicable to any family of orthogonal polynomials, in particular the classical orthogonal polynomials.

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BibTeX

@article {MR2832754,
AUTHOR={Costas-Santos, R. S. and Sanchez-Lara, J. F.},
TITLE={Orthogonality of {$q$}-polynomials for non-standard parameters},
JOURNAL={J. Approx. Theory},
FJOURNAL={Journal of Approximation Theory},
VOLUME={163},
YEAR={2011},
NUMBER={9},
PAGES={1246--1268},
ISSN={0021-9045},
CODEN={JAXTAZ},
MRCLASS={33D45},
MRNUMBER={2832754 (2012f:33027)},
ZBL={1229.33016},
MRREVIEWER={Ulrich Tamm},
DOI={10.1016/j.jat.2011.04.005},
URL={http://dx.doi.org/10.1016/j.jat.2011.04.005},
}