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

Abstract

We describe a new connection between the dilogarithm function and the solutions of 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 $$x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]$$ . We further show that Ramanujan’s dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.