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

2019/03/10 by Martin Bridgeman, Bridgeman, Martin
Mathematics · Physics and Astronomy · #11D09 #51N25 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1903.04074

openalex publication_date 2019/03/10 · openalex created_date 2022/07/24 · openalex updated_date 2026/07/28

Abstract

In this paper we give describe a new connection between the dilogarithm\nfunction and solutions to Pell's equation x²-ny² = \± 1. For each solution\nx,y to Pell's equation we obtain a dilogarithm identity whose terms are given\nby the continued fraction expansion of the associated unit x+y\√(n) \∈\n Z[\√(n)]. We further show that Ramanujan's dilogarithm value-identities\ncorrespond to an identity for the regular ideal hyperbolic hexagon.\n

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