A distributional study of discrete classical orthogonal polynomials. (English) Zbl 0853.33009

García, A. G.; Marcellán, F.; Salto, L.

J. Comput. Appl. Math. 57, No. 1-2, 147-162 (1995).

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. F. Marcellán, A. Branquinho and 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.

Reviewer: M.G.de Bruin (Delft)

MSC:

33C45

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

difference equations; moment functionals

Citations:

Zbl 0793.33009

References:

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