Congruences concerning Legendre polynomials
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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.References
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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
- MR Author ID: 318137
- Email: szh6174@yahoo.com
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2775368