Quadratic Irrationals, Generating Functions, and Lévy Constants

Abstract

We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. Consequently we can compute the Lévy constant of any quadratic irrational explicitly in terms of a finite number of its convergents.

Section 1 (Introduction) and Section 3 rewritten. Some minor corrections. 5 figures added. 29 pages