Orthogonal Polynomials and Continued Fractions from Euler's Point of View

A quick confession: When I teach a course in elementary number theory, I tend to not spend much time (if any) on continued fractions. I have always thought that the topic has a handful of neat results, but it has rarely captured my heart as much as many other topics in the course, and I have never felt like there was enough depth to justify more than a day or two on the topic. If this describes you as well, then I recommend checking out Sergey Khrushchev's Orthogonal Polynomials and Continued Fractions, as you will never question the depth and beauty of continued fractions again.

In the preface to the book, Khrushchev writes that the project "emerged as a result of my attempts to understand the theory of orthogonal polynomials." He writes about how his early exposure to the topic taught him quite a bit but "the fundamental reasons for its beauty and difficulty remained unclear." Several decades later he began once again exploring the topic, partly inspired by reading the original writings of Euler, and this inspired the book under review.

But before he gets to the topic of orthogonal polynomials, he must first introduce continued fractions. He does so in an initial chapter which quickly covers all of the topics that one typically finds in a book on elementary number theory (the same topics I tend to skip in my classes) before going on to give an introduction to Jean Bernoulli sequences and Markoff sequences.

While this initial chapter keeps things (relatively) elementary, the difficulty then begins to pick up steam. The second and third chapter of the book also deal with continued fractions, with chapter two covering algebraic proofs that come out of Markoff's theory and chapter three looks at things from an analytic point of view, culminating in Brouncker's extension of the unit circle method. Euler's work on continued fractions and, in particular, the differential method of summation, takes up the bulk of chapter four, while chapter five collects results which Khrushchev considers in the spirit of Euler. In particular, this chapter gives a proof of the irrationality of π.

The sixth chapter of the book moves from regular continued fractions to polynomial continued fractions, also known as P-fractions. This chapter includes results on hypergeometric functions and periodicity, and also dicusses Euler's substitutions for integration arising from the radicals of quadratic polynomials. This work eventually led Euler to try to parametrize all orthogonal matrices, which in turn led to the consideration of orthogonal polynomials, as explored in chapter seven. The final chapter of the book is based on Khrushchev's own original work on orthogonal polynomials on the unit circle and their connections with continued fractions. The book closes by including a translation of Euler's paper "Continued fractions, observations."

The mathematics contained in this book is beautiful, but it is also difficult, and Khrushchev often does not give as much background or motivation as one might like. However, he has done an admirable job of weaving together historical anecdotes and excerpts from original sources with some deep and modern mathematics as well as some autobiographical tales of his own. The resulting book is a pleasure to read for people interested in either the topic of orthogonal polynomials and continued fractions or for historians of mathematics, and I imagine that any reader will walk away with a deeper appreciation of both.

Darren Glass is an Assistant Professor of Mathematics at Gettysburg College whose research interests include Number Theory, Algebraic Geometry, and Cryptography. He can be reached at dglass@gettysburg.edu.