Mattarei, Sandro and Tauraso, Roberto (2018) From generating series to polynomial congruences. Journal of Number Theory, 182 . pp. 179205. ISSN 0022314X
Full content URL: http://dx.doi.org/10.1016/j.jnt.2017.06.007
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Item Type:  Article 

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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 $\sum_{k=0}^{q1}c_kx^k$, 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 $\sum_{k=0}^{q1}\binom{2k}{k}x^k\equiv(14x)^{(q1)/2}\pmod{p}$ being a finite match for the wellknown generating function $\sum_{k=0}^\infty\binom{2k}{k}x^k= 1/\sqrt{14x}$. We develop a method which allows one to directly infer the closedform 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 closedform 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 closedforms modulo $p$ for the corresponding truncated sums, in terms of finite polylogarithms $\pounds_d(x)=\sum_{k=1}^{p1}x^k/k^d$.
Keywords:  binomial coeffcients, harmonic numbers, polylogarithms, generating functions 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  28782 
Deposited On:  29 Sep 2017 08:55 
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