Une théorie algebrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. (French) Zbl 0944.33500

Maroni, P.

Brezinski, Claude (ed.) et al., Orthogonal polynomials and their applications. Proceedings of the third international symposium held in Erice, Italy, June 1-8, 1990. Basel: J. C. Baltzer, IMACS Ann. Comput. Appl. Math. 9, 95-130 (1991).

From the introduction: (translated from the French): “Since the works of J. Shohat [Duke Math. J. 5, 401-417 (1939; Zbl 0021.30802)], the inventor of semiclassical polynomials, numerous authors have tried to generalize the properties of classical polynomials, generally in terms of their hypergeometric character. In the face of the accumulation of what have often seemed to be unrelated results, the need for structure and classification has naturally arisen. The present study therefore seeks to outline a synthesis of the properties of what are known today as semiclassical orthogonal polynomials; it extends two previous papers of ours [in Orthogonal polynomials and their applications (Segovia, 1986), 279-290 (1988; Zbl 0661.42015); “Polynômes orthogonaux semi-classiques et equations differentielles”, Publ. Lab. Anal. Numer. No. 90002, Univ. Paris VI–CNRS, Paris, 1990; per bibl.].
“The basic idea in this paper consists of working directly with linear forms rather than integral representations, in other words, of pursuing the intrinsic relations that may exist between the forms under consideration, either in the vector space of polynomial functions or in the vector space of formal power series isomorphic to it. Thus, in this approach, we systematically use the dual sequence of a (free) sequence of polynomials”.
For the entire collection see [Zbl 0812.00027].

MSC:

33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Citations:

Zbl 0021.30802; Zbl 0661.42015