, Volume 1, Issue 4, pp 431-448

Algebraic Dilogarithm Identities

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Abstract

The Rogers L-function \(L(x) = \sum\limits_{n = 1}^\infty {\frac{{x^n }} {{n^2 }} + \frac{1} {2}\log x} \log (1 - x) \) satisfies the functional equation \(L(x) + L(y) = L(xy) + L\left( {\frac{{x(1 - y)}} {{1 - xy}}} \right) + L\left( {\frac{{y(1 - x)}} {{1 - xy}}} \right) \) .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form \(\sum\limits_{k = 0}^n {c_k L(\theta ^k ) = 0} \) where θ is algebraic and the c k are integers.