2017/03/07 by Mattarei, Sandro, Tauraso, Roberto · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1703.02322
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\binom2kkxk≡(1-4x)(q-1)/2\pmodp being a finite match for the well-known generating function ∑k=0^∞\binom2kkxk= 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 \rm 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.