New paper
Sandro Mattarei and Roberto Tauraso, From generating series to polynomial congruences, 24 pages, submitted, 2017; arXiv:1703.02322.
Abstract: Consider an ordinary generating function 
, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form 
. Various instances are known where the corresponding truncated sum 
, with 
a power of a prime 
, also admits a closed form representation when viewed modulo . Such a representation for the truncated sum modulo frequently bears a resemblance with the shape of , despite being typically proved through independent arguments. One of the simplest examples is the congruence 
being a finite match for the well-known generating function 
.
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 
, and after supplementing them with some new ones we obtain closed-forms modulo for the corresponding truncated sums, in terms of finite polylogarithms 
.
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