Dilogarithm identities for solutions to Pell’s equation in terms of continued fraction convergents
2020
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.
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