Legendre polynomials and supercongruences

Zhi-Hong Sun. "Legendre polynomials and supercongruences." Acta Arithmetica 159.2 (2013): 169-200. <http://eudml.org/doc/286585>.

@article{Zhi2013,
abstract = {Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_\{[p/6]\}(t) ≡ -(3/p)∑_\{x=0\}^\{p-1\} ((x³-3x+2t)/p) (mod p)$ and $(∑_\{x=0\}^\{p-1\} ((x³+mx+n)/p))² ≡ ((-3m)/p) ∑_\{k=0\}^\{[p/6]\} \binom\{2k\}\{k\}\binom\{3k\}\{k\}\binom\{6k\}\{3k\} ((4m³+27n²)/(12³·4m³))^k (mod p)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning $∑_\{k=0\}^\{p-1\}\binom\{2k\}\{k\}\binom\{3k\}\{k\}\binom\{6k\}\{3k\}/m^\{k\} (mod p²)$, where m is an integer not divisible by p.},
author = {Zhi-Hong Sun},
journal = {Acta Arithmetica},
keywords = {binomial coefficient; congruence; Legendre polynomial; character sum; binary quadratic form},
language = {eng},
number = {2},
pages = {169-200},
title = {Legendre polynomials and supercongruences},
url = {http://eudml.org/doc/286585},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Zhi-Hong Sun
TI - Legendre polynomials and supercongruences
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 2
SP - 169
EP - 200
AB - Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}(t) ≡ -(3/p)∑_{x=0}^{p-1} ((x³-3x+2t)/p) (mod p)$ and $(∑_{x=0}^{p-1} ((x³+mx+n)/p))² ≡ ((-3m)/p) ∑_{k=0}^{[p/6]} \binom{2k}{k}\binom{3k}{k}\binom{6k}{3k} ((4m³+27n²)/(12³·4m³))^k (mod p)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning $∑_{k=0}^{p-1}\binom{2k}{k}\binom{3k}{k}\binom{6k}{3k}/m^{k} (mod p²)$, where m is an integer not divisible by p.
LA - eng
KW - binomial coefficient; congruence; Legendre polynomial; character sum; binary quadratic form
UR - http://eudml.org/doc/286585
ER -