Congruences concerning Legendre polynomials II

Abstract

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning ${\sum }_{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3/m^k(\mathrm{mod}{p}^2)$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that ${\sum }_{k=0}^{\frac{p-1}{2}}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3\equiv 0(\mathrm{mod}{p}^2)$ for $p\equiv 3,5,6(\mathrm{mod}7)$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $P_{[\frac{p}{4}]}(t)\equiv -\left(\frac{-6}{p}\right){\sum }_{x=0}^{p-1}\left(\frac{x^3-3/2(3t+5)x-9t-7}{p}\right)(\mathrm{mod}p)$ and determine $P_{\frac{p-1}{2}}(\sqrt{2}),P_{\frac{p-1}{2}}(\frac{3\sqrt{2}}{4}),P_{\frac{p-1}{2}}(\sqrt{-3}),P_{\frac{p-1}{2}}(\frac{\sqrt{3}}{2}),P_{\frac{p-1}{2}}(\sqrt{-63}),P_{\frac{p-1}{2}}(\frac{3\sqrt{7}}{8})(\mathrm{mod}p)$, where $t$ is a rational $p-$integer, $[x]$ is the greatest integer not exceeding $x$ and $(\frac{a}{p})$ is the Legendre symbol. As consequences we determine $P_{[\frac{p}{4}]}(t)(\mathrm{mod}p)$ in the cases $t=-5/3,-7/9,-65/63$ and confirm many conjectures of Z.W. Sun.

25 pages. arXiv admin note: text overlap with arXiv:1104.3047, arXiv:1104.2789