The \(I_{(q,\omega)}\) classical orthogonal polynomials
The aim of the paper is to characterize the orthogonal polynomials with respect a linear functional whose moments satisfy a recurrence relation of the form \[ (u)_{n+1}=\frac{a+bq^{n+1}}{c+dq^{n+1}}(u)_n,\quad n\geq 0,\tag{1} \] with \(ad-bc\neq 0\). It is shown that this recurrence relation is equivalent to a functional equation in terms of the operator \(I_{(q,\omega)}\) defined by \[ \left(I_{(q,\omega)}f\right)(x):=f(x)+\omega f(qx) \] for any polynomial \(f\). The authors show that the sequence \(\{I_{(q,\omega)}P_n\}_{n\geq 0}\) is orthogonal if and only if the sequence \(\{P_n\}_{n\geq 0}\) is orthogonal with respect to a functional whose moments satisfy (1). These so-called \(I_{(q,\omega)}\)-classical orthogonal polynomials form a part of the well-known Hahn class of orthogonal polynomials. The last section of the paper deals with integral representations for the corresponding moment functionals.
scientific article; zbMATH DE number 1879039