Congruences concerning Legendre polynomials

Details Congruences concerning Legendre polynomials Congruences concerning Legendre polynomials

2010-12-17

arxiv.org/abs/1012.3833v1

Mathematics

Number Theory

16 pages

Scientific paper

10.1090/S0002-9939-2010-10566-X

Let $p$ be an odd prime. In the paper, by using the properties of Legendre
polynomials we prove some congruences for
$\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we
confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on
supercongruences.

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