Volume 236, 1999
Continued fractions and orthogonal polynomials
To Jerry Lange for carrying on the work with continued fractions
Recurrence relations are very important when one studies orthog-
onal polynomials. Examples are given, including some where the recurrence
relations relate to continued fractions.
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
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
years when Szego wrote , with the
work of Legendre and Laplace on spherical harmonics being one of the starting
places. Gauss  developed Gaussian quadrature from continued fractions. How-
ever, Jacobi's version  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