New paper accepted in Journal of Number Theory

The paper by Sandro Mattarei and Roberto Tauraso, From generating series to polynomial congruences, has been accepted for publication in Journal of Number Theory.

(You may find the final version in preprint form at https://arxiv.org/pdf/1703.02322.pdf.)

Abstract: Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$ , 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 $\sum_{k=0}^{q-1}c_kx^k$ , 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 $\sum_{k=0}^{q-1}\binom{2k}{k}x^k\equiv(1-4x)^{(q-1)/2}\pmod{p}$ being a finite match for the well-known generating function $\sum_{k=0}^\infty\binom{2k}{k}x^k= 1/\sqrt{1-4x}$ . 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 ${\rm Li}_d(x)=\sum_{k=1}^{\infty}x^k/k^d$ , and after supplementing them with some new ones we obtain closed-forms modulo $p$ for the corresponding truncated sums, in terms of finite polylogarithms $\pounds_d(x)=\sum_{k=1}^{p-1}x^k/k^d$ .