Legendre polynomials and supercongruences

Zhi-Hong Sun¹

¹ School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 223001, P.R. China

Acta Arithmetica, Tome 159 (2013) no. 2, pp. 169-200

Résumé

Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is not divisible by $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\equiv 0\pmod p$, $$ P_{[\frac p6]}(t) \equiv -\biggl(\frac 3p\biggr)\sum_{x=0}^{p-1}\biggl(\frac{x^3-3x+2t}p\biggr)\pmod p $$ and $$ \biggl(\sum_{x=0}^{p-1}\biggl(\frac{x^3+mx+n}p\biggr)\biggr)^2 \equiv \biggl(\frac{-3m}p\biggr) \sum_{k=0}^{[p/6]}\binom{2k}k\binom{3k}k\binom{6k}{3k} \biggl(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\biggr)^k\pmod p, $$ where $(\frac ap)$ 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 $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k\binom{6k}{3k}/m^k\pmod {p^2}$, where $m$ is an integer not divisible by $p$.

Détail

DOI : 10.4064/aa159-2-6

Mots-clés : prime set rational numbers whose denominator divisible legendre polynomials paper mainly equiv pmod frac equiv biggl frac biggr sum p biggl frac biggr pmod biggl sum p biggl frac biggr biggr equiv biggl frac biggr sum binom binom binom biggl frac cdot biggr pmod where frac legendre symbol greatest integer function application solve conjectures sun author concerning sum p binom binom binom pmod where integer divisible nbsp