Partially-orthogonal polynomials

Abstract

This paper contains a discussion of partiallyorthogonal polynomials. This is an extension of the concept of quasi-orthogonal polynomials. Some relationships between various partially-orthogonal polynomials are obtained. The concept of pseudo-polynomials is defined and used as an example of partially-orthogonal polynomials. Polynomials obtained from the simple Laguerre polynomials are also used as an example. The concept of quasi-orthogonal polynomials is discussed by Dickinson [2] and by Chihara [1]. It is the purpose of this paper to discuss some generalizations of the concept of quasi-orthogonal polynomials and to obtain recurrence relations between the various polynomials. Some examples will be given. Also, the concept of polynomials will be generalized. DEFINITION 1. Let {Qn(x, m)}l=0 be a set of polynomials, where each Qn(x, m) is of degree n. The Qn(x, m) will be called partially-orthogonal of deficiency m if there exists an interval (a, b) such that fw(x)x'Qk(x,m)dx=O forO m, $AO forj>k-m,k>m, where w(x) is a nonnegative weight function. If m=O the set of polynomials are fully orthogonal. If m=1 the set of polynomials are quasiorthogonal. The m=O index will be omitted in this paper. For simplicity all the examples of polynomials here will have leading coefficient unity, and this is assumed throughout the paper. DEFINITION 2. We will call two partially-orthogonal sets of polynomials related if the weight function and interval of integration are the same but the deficiencies are different. DEFINITION 3. Two polynomials will be said to share the same zero if they are both annihilated by the same operation. By operation is meant any linear functional F, and F annihilates the polynomial Q if F(Q)=O. Received by the editors February 1, 1971. AMS 1970 subject classifications. Primary 33A65.