A refinement of a congruence result by van Hamme and Mortenson. (English) Zbl 1292.11040

Sun, Zhi-Wei

Ill. J. Math. 56, No. 3, 967-979 (2012).

Summary: Let \(p\) be an odd prime. In 2008, E. Mortenson [Proc. Am. Math. Soc. 136, No. 12, 4321–4328 (2008; Zbl 1171.11061)] proved van Hamme’s following conjecture: \[ \sum^{(p-1)/2}_{k=0} (4k+1)\binom{-1/2}{k}^3\equiv(-1)^{(p-1)/2}p\pmod {p^3}. \] In this paper, we show further that \[ \begin{aligned} \sum^{(p-1)}_{k=0}(4k+1)\binom{-1/2}{k}^3 & \equiv \sum^{(p-1)/2}_{k=0}(4k+1)\binom{-1/2}{k}^3 \\ & \equiv (-1)^{(p-1)/2}p+p^3E_{p-3}\pmod {p^4},\end{aligned} \] where \(E_{0},E_{1},E_{2},\ldots \) are Euler numbers. We also prove that if \(p>3\) then \[ \sum^{(p-1)/2}_{k=0}\frac{20k+3}{(-2^{10})^{k}}\binom{4k}{k,k,k,k} \equiv (-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2) \pmod {p^4}. \]

MSC:

11B65	Binomial coefficients; factorials; \(q\)-identities
05A10	Factorials, binomial coefficients, combinatorial functions
11A07	Congruences; primitive roots; residue systems
11B68	Bernoulli and Euler numbers and polynomials

Citations:

Zbl 1171.11061

References:

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