A distributional study of discrete classical orthogonal polynomials
This paper is a systematic study of discrete classical orthogonal polynomial sequences (Charlier, Meixner, Hahn) from a distributional viewpoint. The properties for the continuous case orthogonal polynomial sequences (denoted here by DOPS; Hermite, Laguerre, Jacobi, Bessel) can a.o. be derived from a distributional differential equation \(D (\varphi u) = \psi u\) for the associated linear functional \(u\) (cf. \textit{F. Marcellán}, \textit{A. Branquinho} and \textit{J. Petronilho} [Acta Appl. Math. 34, 283-303 (1994; Zbl 0793.33009)]). The authors replace the differential operator \(D\) and the word `differential' by the difference operator \(\Delta\) and the word `difference' and derive properties analogous to those in the continuous case (Sturm-Liouville differential equation of order 2, \(D(P_n)\) again is a DOPS, Rodrigues formula, structure relation, Pearson differential equation for the associated weight). As stated in the opening sentence of this review the work is done systematically: in a (large) number of small steps (4 definitions, 10 lemmas, 9 propositions, 1 theorem and several remarks) the results are derived. A well written paper, suitable for inclusion in a (graduate) textbook on orthogonal polynomials.
scientific article; zbMATH DE number 770342