An introduction to the \(H_q\)-semiclassical orthogonal polynomials
The study of \(q\)-polynomials has known an increasing interest in the last years, specially the Askey-Wilson polynomials and the polynomials related to them. In this paper the author considers the so-called \(q\)-semiclassical polynomials, i.e., an extension of the classical \(q\)-polynomials [see e.g. \textit{J. C. Medem, R. Álvarez-Nodarse}, and \textit{F. Marcellán}, J. Comput. Appl. Math. 135, No. 2, 157--196 (2001; Zbl 0991.33007)] introduced by Hahn in 1949. The approach is based on the distributional \(q\)-Pearson equation \[ H_q(\Phi u)+\Psi u=0, \] where \(\Phi,\Psi\) are polynomials of degree \(t\) and \(p\) such that the leading coefficient of \(\Psi\) is different from \([n+1]=(q^{n+1}-1)/(q-1)\) (i.e., \((\Phi,\Psi)\) is an admissible pair). Using a technique introduced and developed by Maroni the author establish several important properties such as the class of the \(H_q\) functional \(u\). In particular, several characterizations are proved among which a second-order linear difference equation with polynomial coefficients (that depends also on \(n\), the degree of the polynomials) is obtained. Finally, an example of \(q\)-semiclassical polynomials of class 1 is considered in details. Some of the results presented here already appeared in the Doctoral dissertation by J. C. Medem in 1996.
\(q\)-semiclassical orthogonal polynomials
scientific article; zbMATH DE number 2110024