Chain sequences and orthogonal polynomials

Abstract

and the above results also follow from the classical theory of continued fractions (for example, see [10] and the remarks in [5]). In view of this, it seems natural to attempt a study of the properties of the polynomials, Pn(x), as determined by the coefficients, cn and Xn. One approach to such an investigation is of course through the study of the convergence of the continued fraction (1.3). In this paper, however, we shall avoid direct reference to the theory of continued fractions although we shall make fairly extensive use of the results from the problem of moments. Instead, our investigation centers upon the true interval of orthogonality of Pn(x) }. [By the true interval of orthogonality, we mean, following Shohat (see [5; 6, p. 113]), the smallest interval which contains in its interior all zeros of all Pn(x).] In this connection, the chain sequences of Wall [10] arise naturally and play a fundamental role. Thus our study is largely arithmetical in character. Throughout this paper, orthogonal means polynomials