Legendre polynomials and supercongruences

• Autorzy: Zhi-Hong Sun
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11L10: Jacobsthal and Brewer sums; other complete character sums
• 05A19: Combinatorial identities, bijective combinatorics
• 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
• 05A10: Factorials, binomial coefficients, combinatorial functions
• 11E25: Sums of squares and representations by other particular quadratic forms
• 11A07: Congruences; primitive roots; residue systems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 2
• Strony: 169-200

Daty

wydano

2013

Twórcy

autor

Zhi-Hong Sun

• School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 223001, P.R. China

Identyfikatory

DOI

10.4064/aa159-2-6