From generating series to polynomial congruences

JOURNAL OF NUMBER THEORY

卷:182

Article

Mattarei, Sandro¹  Tauraso, Roberto² 

[1] Univ Lincoln, Sch Math & Phys, Lincoln LN6 7TS, England

[2] Univ Roma Tor Vergata, Dipartimento Matemat, Via Ric Sci, I-00133 Rome, Italy

关键词: Congruences;    Binomial coefficients;    Harmonic numbers;    Polylogarithms;    Generating functions;   

DOI  :  10.1016/j.jnt.2017.06.007

来源: Elsevier

【 摘 要 】

Consider an ordinary generating function Sigma(infinity)(k-0) CkXk, 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 Sigma(q-1)(k-0) c(k)x(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 Sigma(q-1)(k-0) ((2k)(k))x(kappa) equivalent to (1 - 4x)((q-1)/2) (mod p) being a finite match for the well-known generating function Sigma(infinity)(k-0) ((2k)(k))x(kappa) = 1/root 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 Li-d (x) = Sigma(infinity)(k-1) 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 (d)(x) pound = Sigma(p-1)(k=1) x(k)/k(d). (C) 2017 Elsevier Inc. All rights reserved.

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