Congruences concerning Legendre polynomials
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- 2 November 2010
- journal article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 139 (6) , 1915-1929
Abstract
Let p p be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for ∑ k = 0 p − 1 2 ( 2 k k ) 2 m − k ( mod p 2 ) \sum _{k=0}^{\frac {p-1}2}\binom {2k}k^{2}m^{-k}(\textrm {mod}\, {p^{2})} . In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.Keywords
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