From generating series to polynomial congruences

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 Lid(x)=∑k=1∞xk/kd, and after supplementing them with some new ones we obtain closed-forms modulo p for the corresponding truncated sums, in terms of finite polylogarithms £d(x)=∑k=1p−1xk/kd.