A dilogarithm identity?

Question

I'm wondering whether there any nice identities (or relationships) that can simplify or possibly compact the following expressions:

$$\operatorname{Li}_2(\beta e^{\alpha x}) - \operatorname{Li}_2(\beta e^{-\alpha x}) $$

and

$$\operatorname{Li}_2(pe^{\alpha x}) + \operatorname{Li}_2(qe^{\alpha x}) $$

Answer 1

Note that

$$\operatorname{Li}_2(\beta e^{ix})-\operatorname{Li}_2(\beta e^{-ix})=\sum_{k=1}^\infty\frac{\beta^k}{k^2}(e^{ikx}-e^{-ikx})=2i\sum_{k=1}^\infty\frac{\sin(kx)}{k^2}\beta^k$$

Assuming $\beta$ and $x$ are real, we may use Euler's formula to deduce that

$$\operatorname{Li}_2(\beta e^{ix})-\operatorname{Li}_2(\beta e^{-ix})=-2\Re(\operatorname{Li}_2(\beta e^{ix}))=2\Re(\operatorname{Li}_2(\beta e^{-ix}))$$

And I don't see any hope for anything better for this one or the other.