Congruences concerning Legendre polynomials

Sun, Zhi-Hong

doi:10.1090/S0002-9939-2010-10566-X

Congruences concerning Legendre polynomials
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Proc. Amer. Math. Soc. 139 (2011), 1915-1929 Request permission

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

Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. MR 1739404, DOI 10.1515/crll.2000.004
Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
S. Chowla, B. Dwork, and Ronald Evans, On the mod $p^2$ determination of $\left ({(p-1)/2\atop (p-1)/4}\right )$, J. Number Theory 24 (1986), no. 2, 188–196. MR 863654, DOI 10.1016/0022-314X(86)90102-2
Henry W. Gould, Combinatorial identities, Henry W. Gould, Morgantown, W. Va., 1972. A standardized set of tables listing 500 binomial coefficient summations. MR 0354401
Po-Ru Loh and Robert C. Rhoades, $p$-adic and combinatorial properties of modular form coefficients, Int. J. Number Theory 2 (2006), no. 2, 305–328. MR 2240232, DOI 10.1142/S1793042106000590
Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
Eric Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), no. 1, 139–147. MR 1957248, DOI 10.1016/S0022-314X(02)00052-5
Eric Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), no. 3, 987–1007. MR 1938742, DOI 10.1090/S0002-9947-02-03172-0
Eric Mortenson, Supercongruences for truncated $_{n+1}\!F_n$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), no. 2, 321–330. MR 2093051, DOI 10.1090/S0002-9939-04-07697-X
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
Fernando Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 223–231. MR 2019156
Zhi-Hong Sun, On the number of incongruent residues of $x^4+ax^2+bx$ modulo $p$, J. Number Theory 119 (2006), no. 2, 210–241. MR 2250045, DOI 10.1016/j.jnt.2005.10.012
Z.W. Sun, On congruences related to central binomial coefficients, preprint, arXiv:0911.2415, http://arxiv.org/abs/0911.2415.
Z.W. Sun, Open conjectures on congruences, arXiv:0911.5665. http://arxiv.org/abs/ 0911.5665.
L. Van Hamme, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference on $p$-adic analysis (Houthalen, 1987) Vrije Univ. Brussel, Brussels, 1986, pp. 189–195. MR 921871