Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients

Abstract

We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients $\alpha_n$ and $\beta_n$ in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of $n$ and are determined by a set of six parameters. The partial fraction decomposition of $\alpha_n$ is a linear combination of five linearly independent functions of $n$ with coefficients $d_j$, for $0 \le j \le 4$, that are polynomials in our six parameters. For each value of a parameter $r$, associated with the eigenvalues, we classify the $\alpha_n$ according to the set of coefficients $d_j$ that are nonzero. There are certain particular values of $r$ that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.