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

Martin Bridgeman

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arXiv · 1903.04074

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

Martin Bridgeman

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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.

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Martin Bridgeman. 2019-03-10. Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents. https://arxiv.org/abs/1903.04074

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