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From generating series to polynomial congruences | | Sandro Mattarei
; Roberto Tauraso
; | | Date: |
7 Mar 2017 | | 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
$sum_{k=0}^{q-1}inom{2k}{k}x^kequiv(1-4x)^{(q-1)/2}pmod{p}$ being a finite
match for the well-known generating function
$sum_{k=0}^inftyinom{2k}{k}x^k= 1/sqrt{1-4x}$. 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
${
m Li}_d(x)=sum_{k=1}^{infty}x^k/k^d$, and after supplementing them with
some new ones we obtain closed-forms modulo $p$ for the corresponding truncated
sums, in terms of finite polylogarithms $pounds_d(x)=sum_{k=1}^{p-1}x^k/k^d$. | | Source: | arXiv, 1703.2322 | | Services: | Forum | Review | PDF | Favorites | |
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