Contemporary Mathematics

Volume 236, 1999

Continued fractions and orthogonal polynomials

Richard Askey

To Jerry Lange for carrying on the work with continued fractions

ABSTRACT.

Recurrence relations are very important when one studies orthog-

onal polynomials. Examples are given, including some where the recurrence

relations relate to continued fractions.

1.

Introduction

Gabor Szego wrote the following in the preface of his great book "Orthogonal

Polynomials", [34, p. v].

"The origins of the subject [orthogonal polynomials] are to be

found in the investigation of a certain type of continued fractions,

bearing the name of Stieltjes, ... Despite the close relationship

between continued fractions and the problem of moments, and

notwithstanding recent important advances in this latter subject,

continued fractions have been gradually abandoned as a starting

point for the theory of orthogonal polynomials."

When Szego wrote his book sixty years ago, it was not clear what role was

left for continued fractions. There are two dominant themes in Szego's book: the

classical orthogonal polynomials of Jacobi, Laguerre and Hermite, and the general

theory of polynomials orthogonal on a bounded interval, say

[-1,1]'

whose weight

function is positive in the interior of this interval and does not vanish too rapidly

as the variable approaches the endpoints of this interval. The classical orthogonal

polynomials had been studied for about

150

years when Szego wrote [34], with the

work of Legendre and Laplace on spherical harmonics being one of the starting

places. Gauss [14] developed Gaussian quadrature from continued fractions. How-

ever, Jacobi's version [18] of this using orthogonality was much easier to motivate,

and to extend to other measures.

Szego's theory of orthogonal polynomials on an interval, and the corresponding

theory on the unit circle, were developed independently from continued fractions.

1991 Mathematics Subject Classification.

Primary 33C45, 33D45; Secondary 30870.

Key words and phrases.

Orthogonal polynomials, continued fractions, recurrence relations.

Supported in part by a grant from the University of Wisconsin Graduate School.

©

1999 American Mathematical Society

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http://dx.doi.org/10.1090/conm/236/03487