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

## TL;DR

This paper establishes a novel link between the dilogarithm function and solutions to Pell's equation, using continued fraction expansions, and connects Ramanujan's identities to hyperbolic geometry.

## Contribution

It introduces a new method to derive dilogarithm identities from Pell's equation solutions via continued fractions, and relates Ramanujan's identities to hyperbolic geometry.

## Key findings

- Dilogarithm identities associated with Pell's solutions are derived.
- Ramanujan's dilogarithm identities are interpreted through hyperbolic geometry.
- Continued fraction expansions of units in quadratic fields are central to the identities.

## Abstract

In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-ny^2 = \pm 1$. 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 $x+y\sqrt{n} \in \Z[\sqrt{n}]$. We further show that Ramanujan's dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.