Quadratic Irrationals, Generating Functions, and L\'evy Constants

TL;DR

This paper demonstrates that the generating function for the denominators of best rational approximations of quadratic irrationals is rational with integer coefficients, enabling explicit computation of their Le9vy constants.

Contribution

It establishes the rationality of the generating function for quadratic irrationals' approximants and provides a method to compute their Le9vy constants explicitly.

Findings

Generating function is a rational function with integer coefficients.

Le9vy constants can be computed explicitly from finitely many convergents.

Provides a new approach to analyze quadratic irrationals' approximation properties.

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 L\'evy constant of any quadratic irrational explicitly in terms of a finite number of its convergents.