Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents

Abstract

In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-n{y}^2=\pm 1$. For each solution $x,y$ to Pell's equation we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $x+y\sqrt{n}\in \mathbb{Z}[\sqrt{n}]$. We further show that Ramanujan's dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.

15 pages, 2 figures