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 \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 \mathrm{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.


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