Analytic Dilogarithm Identities

2025/06/11 by Cetin Hakimoglu-Brown, Hakimoglu-Brown, Cetin
Mathematics · #33B15 #33B30 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.10206

openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce dilogarithm identities through a beta integral-based technique that we apply to provide analytic proofs of previously conjectured dilogarithm relations, solving open problems given by both Bytsko and Campbell, and that we further apply to construct and prove new ladder relations with quartic and sextic bases. We also apply our method to introduce and prove two-term Li₂-relations and ladder-like identities with arguments in algebraic number fields such as ℚ(√(2)), ℚ(√(3)), ℚ(√(5)), and ℚ(√(-7)). Moreover, single-term dilogarithm evaluations are introduced and derived.

Citations

• On the minimal polynomials of the arguments of dilogarithm ladders
• Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for π and Values of the Gamma Function
• New series involving binomial coefficients (II)
• A series of Ramanujan, two-term dilogarithm identities and some Lucas series
• Conformal Field Theory and Torsion Elements of the Bloch Group
• Two-term dilogarithm identities related to conformal field theory

Related

• Functional Dilogarithm Identities in Quadratic Fields
• On the minimal polynomials of the arguments of dilogarithm ladders
• Dilogarithm identities after Bridgeman
• Families of self-inverse functions and dilogarithm identities
• Wild quantum dilogarithm identities