Charlotte Scott Centre for Algebra

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 $latex sum_{k=0}^{infty}c_kx^k$, 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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