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Congruences concerning Legendre polynomials


Author: Zhi-Hong Sun
Journal: Proc. Amer. Math. Soc. 139 (2011), 1915-1929
MSC (2010): Primary 11A07; Secondary 33C45, 11E25
DOI: https://doi.org/10.1090/S0002-9939-2010-10566-X
Published electronically: November 2, 2010
MathSciNet review: 2775368
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Abstract: Let $ p$ be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for $ \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.


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Additional Information

Zhi-Hong Sun
Affiliation: School of the Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 223001, People’s Republic of China
Email: szh6174@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-2010-10566-X
Keywords: Legendre polynomial, congruence
Received by editor(s): November 23, 2009
Received by editor(s) in revised form: May 27, 2010
Published electronically: November 2, 2010
Additional Notes: The author is supported by the Natural Sciences Foundation of China (grant No. 10971078)
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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