Une caractérisation des polynômes strictement 1/p orthogonaux de type Scheffer. Étude du cas \(p=2\). (A characterization of strictly 1/p orthogonal polynomials of Scheffer type) (Q1113410)

A sequence \(\{P_ n(x)\}\), \(n\geq 0\), of polynomials is called of type A zero if it is defined by means of a generating function of the form \(G(x,t)=A(t)e^{xH(t)}=\sum_{n\geq 0}P_ n(x)t^ n/n!\). Sequences of such type are considered, which are strictly 1/p orthogonal (p\(\geq 2\) fixed) and a characterization is given by means of two differential equations satisfied by A and H respectively. In only one of the two cases which arise the sequence of polynomials \(P_ n\) satisfy a \(p+1\)-term recurrence relation. With the results obtained it is shown that there are exactly nine families of strictly semi-orthogonal polynomials satisfying three term recurrence relations and five families which satisfy four term recurrence relations.