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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
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