Congruences concerning Legendre polynomials. II.(English)Zbl 1277.11002

Let $$p>3$$ be a prime, and let $$m$$ be an integer with $$p\nmid m$$. In this paper the author solves some conjectures of Z. W. Sun concerning $\sum_{k=0}^{p-1}{2k\choose k}^3/m^k\pmod{p^2},\qquad \sum_{k=0}^{p-1}{2k\choose k}{4k\choose 2k}/m^k\pmod{p^2}$ and $\sum_{k=0}^{p-1}{2k\choose k}^2{4k\choose 2k}/m^k\pmod{p^2}.$ In particular, the author shows that $$\sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\equiv 0\pmod{p^2}$$ for $$p\equiv 3, 5, 6\pmod 7$$. The author also showed some congruences of Legendre polynomials modulo $$p$$ and confirmed many conjectures of Z. W. Sun.
Part I, see Proc. Am. Math. Soc. 139, No. 6, 1915–1929 (2011; Zbl 1225.11006),

MSC:

 11A07 Congruences; primitive roots; residue systems 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 11E25 Sums of squares and representations by other particular quadratic forms

Citations:

Zbl 1225.11006; Zbl 1277.11003
Full Text: