From generating series to polynomial congruences

Abstract

Consider an ordinary generating function ∑k=0∞ckxk, 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 ∑k=0q−1ckxk, 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 ∑k=0q−1(2kk)xk≡(1−4x)(q−1)/2(modp) being a finite match for the well-known generating function ∑k=0∞(2kk)xk=1/1−4x.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.