Congruences concerning Legendre polynomials III

Details Congruences concerning Legendre polynomials III Congruences concerning Legendre polynomials III

2010-12-20

arxiv.org/abs/1012.4234v3

Mathematics

Number Theory

29 pages

Scientific paper

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 $\sum_{k=0}^{p-1}\frac{(6k)!}{m^k(3k)!k!^3}\mod p$, and show that for integers $m,n$ with $p\nmid m$, $$\Big(\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big)\Big)^2\e \Big(\frac{-3m}p\Big) \sum_{k=0}^{[p/6]}\frac{(6k)!}{(3k)!k!^3}\Big(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\Big)^k\mod p,$$ where $(\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer function. Let $\{P_n(x)\}$ be the Legendre polynomials. We also prove congruences for $P_{[\frac p6]}(t)$ and $P_{[\frac p3]}(t)\mod p$ by using character sums and Morton's work.

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