I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to calculate some integral involving a complicated product of logs/polylogs/algebraic functions, like this one, this one, or this one, or an infinite series involving harmonic numbers, like this one or this one. There are a lot of similarities between their answers as well: they usually end up being a linear combination of rational multiples of powers of pi, logarithms of algebraic numbers, integer zeta-function values, and polylogarithm evaluations.
Although I have very little experience/knowledge regarding logarithms, these similarities suggest that there might be a more general algebraic approach to calculating these types of sums/integrals/polylog identities. The extremely general nature of the top-voted answer here is even more suggestive of this possibility - however, although it demonstrates the existence of a general expression in terms of multiple polylogarithm values, it doesn't discuss ways of simplifying these values (which, ideally, would also be part of an algebraic theory of polylogs).
What kind of research exists about the algebraic properties of special functions consisting of, say, products of polylogs and algebraic functions, or polylogs and rational functions, and their derivatives/antiderivatives? Does differential field theory have anything interesting to say about this, for instance?
Is there a canonical reference book or paper which discusses this topic in depth, from an algebraic perspective?
At the bottom of Wolfram's page on polylogs, they refer to an "amazing identity" involving a special value of $\text{Li}_{17}(\alpha_1^{-17})$, where $\alpha_1$ is defined as the root of a polynomial. Are there any general references exploring the general problem of finding linear dependences between polylog values at algebraic arguments? Again, does differential field theory shed any light here?
