Summary: Consider an ordinary generating function \(\sum_{k = 0}^\infty c_k x^k\), of an integer sequence of some combinatorial relevance, and assume that it admits a closed form \(C(x)\). Various instances are known where the corresponding truncated sum \(\sum_{k = 0}^{q - 1} c_k x^k\), with \(q\) a power of a prime \(p\), also admits a closed form representation when viewed modulo \(p\). Such a representation for the truncated sum modulo \(p\) frequently bears a resemblance with the shape of \(C(x)\), despite being typically proved through independent arguments. One of the simplest examples is the congruence \(\sum_{k = 0}^{q - 1}\binom{2 k}{k}x^k \equiv(1 - 4 x)^{(q - 1) / 2}(\operatorname{mod} p)\) being a finite match for the well-known generating function \(\sum_{k = 0}^\infty\binom{2 k}{k} x^k = 1 / \sqrt{1 - 4 x}\).
We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms \(\mathrm{Li}_d(x) = \sum_{k = 1}^\infty x^k / k^d\), and after supplementing them with some new ones we obtain closed-forms modulo \(p\) for the corresponding truncated sums, in terms of finite polylogarithms \(\text{\textsterling}_d(x) = \sum_{k = 1}^{p - 1} x^k / k^d\).