The polylogarithm satisfies the fundamental identities (3) (4) ... by (6) This latter fact provides a remarkable proof of the Wallis Formula. Some strange 3-adic identities (Solution to Problem 6625) Amer. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon. Theorem 6.1). Wadim Zudilin (CARMA, UoN) Rogers{Ramanujan and dilogarithm identities 12{14 December 2012 4 / 17 another approach to the dilogarithm identities (1.8) was proposed and a proof,differentfromours,oftheperiodicity(1.7)andtheidentities(1.8) forthe(A n,A 1)theorieswasoutlined. Then we have f(x)=const L(x) We continue the function L(x) on all real axisR= R1 [f1g by the following rules 1.2. The dilogarithm is implemented in the Wolfram Language as PolyLog[2, z]. INTRODUCTION The main goal of this paper is to establish a specific connection between classical parti-tion combinatorics and the theory of quiver representations. Read "Algebraic Dilogarithm Identities, The Ramanujan Journal" on DeepDyve, the largest online rental service for scholarly research with thousands of … ... (16) and (17) No general Algorithm is know for the integration of polylogarithms of functions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Using this new method, we also give an alternative and simplified proof of the period- Contents. Recently dilogarithm identities have made their appearance in the physics literature. As in [19], we will approach several q-series identities using the quantum dilogarithm ˚(x) := Q i 0 (1 q ix). Relations between crystal bases, branching functions b λ kΛ 0 (q) and Kostka-Foulkes polynomials (Lusztig's q-analog of weight multiplicity) are considered. The corresponding two-term dilogarithm identities (some of which appear to be new) are obtained. Math. Math. We also discuss connections with algebraic K-theory. They are associated with a mutation sequence of a quiver, and fit nicely with the framework of the cluster algebras [ 13 , 14 ] and their quantization [ 11 , 12 ]. Theorem (c.f. Additionally, it is displayed how fermionic character expressions imply dilogarithm identities for the effective central charge of the conformal field theory in question. Quantum dilogarithm 2.1. Phys. Quantum dilogarithm identities have a rich history and connections to partition counting, BPS spectra in quantum field theories, stability conditions for quiver representations, Poincare series of cohomological Hall algebras, cluster algebras/categories, and Donaldson–Thomas invariants. Plotting tan(x)=x to show roots. For the proof of li_2(x) +li_2(-x) =½li_2(x²) rewrite the dilogs at the right hand side as sums We have €(x^n/n²) +((-x) ^n/n²) from 1 to infinity. CRYSTAL BASES, DILOGARITHM IDENTITIES, AND TORSION IN ALGEBRAIC K-THEORY EDWARD FRENKEL AND ANDRAS SZENES 1. The … the dilogarithm identities for the Y-systems of type Br at any level. The polylogarithm identities lead to remarkable expressions. A dilogarithm identity? Proof of Theorem 1.6 20 Acknowledgements 32 References 33 1. Quantum dilogarithm identities for the square product of A-type Dynkin quivers . 47 (2018), no. 1 ) E L(3j) = 6k32 J=1 6k 2 where L(z) is the Rogers dilogarithm function: L(z)= logwdlog(l-w)-2logzlog(l-z), 0

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