New paper accepted in Journal of Number Theory

Algebra in Lincoln

The paper by Sandro Mattarei and Roberto Tauraso, From generating series to polynomial congruenceshas 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 $latex sum_{k=0}^{infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $latex C(x)$. Various instances are known where the corresponding truncated sum $latex sum_{k=0}^{q-1}c_kx^k$, with $latex q$ a power of a prime $latex p$, also admits a closed form representation when viewed modulo $latex p$. Such a representation for the truncated sum modulo $latex p$ frequently bears a resemblance with the shape of $latex C(x),$ despite being typically proved through independent arguments. One of the simplest examples is the congruence $latex sum_{k=0}^{q-1}binom{2k}{k}x^kequiv(1-4x)^{(q-1)/2}pmod{p}$ being a finite match for the well-known generating function $latex sum_{k=0}^inftybinom{2k}{k}x^k= 1/sqrt{1-4x}$. We develop a method which…

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