Dilogarithm identity containing the tribonacci constant

Solution 1:

To simplify the formulas, I will write $\sum_i\operatorname{Li}_2(x_i)\simeq \sum_j\operatorname{Li}_2(y_j)$ if both sides differ by elementary functions of $x_i$, $y_j$. We have in particular \begin{gather*} \operatorname{Li}_2(-x)\simeq -\operatorname{Li}_2(x)+\frac12\operatorname{Li}_2\left( x^2\right),\\ \operatorname{Li}_2(x)\simeq -\operatorname{Li}_2\left( x^{-1}\right)\simeq \operatorname{Li}_2\left( \frac1{1-x}\right), \end{gather*} so that \begin{gather*} \operatorname{Li}_2\left( \frac{\tau}{\tau+1}\right)\simeq \operatorname{Li}_2\left( -\tau^{-1}\right)\simeq -\operatorname{Li}_2\left( \tau^{-1}\right)+\frac12\operatorname{Li}_2\left( \tau^{-2}\right),\\ \operatorname{Li}_2\left( \tau^{2}\right)=-\operatorname{Li}_2\left( \tau^{-2}\right). \end{gather*} The identity we want to prove can therefore be written as $$\tag{$\clubsuit$}\operatorname{Li}_2\left( \sigma^{3}\right)-\frac12 \operatorname{Li}_2\left( \sigma^{2}\right)-\operatorname{Li}_2\left( \sigma\right)\simeq0,$$ where $\sigma=\tau^{-1}$ satisfies the algebraic equation $\sigma^3+\sigma^2+\sigma=1$.

Identities of the form $\sum_{k=1}^N r_k\operatorname{Li}_2\left(\alpha^k\right)\simeq0$ with $r_k\in\mathbb Q$ and algebraic $\alpha$ are called dilogarithmic ladders. There exists a well-developed technology of their discovery and many classification results. A classical reference is the book Structural properties of polylogarithms (ed. L. Lewin). In fact, ($\clubsuit$) is a linear combination of ladders (3.80) and (3.81) given on its page 41.