More congruences for central binomial coefficients. (English) Zbl 1208.11027
Let \(p>5\) be a prime number. The author proves that
\[
\sum_{k=1}^{p-1}\frac{1}{k^2}\binom{2k}{k}^{-1}\equiv\frac13 \frac{H(1)}{p}\pmod {p^3}
\]
and that
\[
\sum_{k=1}^{p-1}\frac{(-1)^k}{k^3}\binom{2k}{k}^{-1}\equiv-\frac25 \frac{H(1)}{p^2}\pmod {p^3},
\]
where \(H(1)=\sum_{k=1}^{p-1}\frac{1}{k}\).
Reviewer: Florin Nicolae (Berlin)
MSC:
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11A07 | Congruences; primitive roots; residue systems |
References:
| [1] | Almkvist, G.; Granville, A., Borwein and Bradley’s Apèry-like formulae for \(\zeta(4 n + 3)\), Experiment. Math., 8, 197-203 (1999) · Zbl 0976.11035 |
| [2] | Apèry, R., Irrationalité de \(\zeta(2)\) et \(\zeta(3)\), Journées arithmétiques de Luminy. Journées arithmétiques de Luminy, Astérisque, 61, 11-13 (1979) · Zbl 0401.10049 |
| [3] | Borwein, J. M.; Bradley, D. M., Empirically determined Apèry-like formulae for \(\zeta(4 n + 3)\), Experiment. Math., 6, 181-194 (1997) · Zbl 0887.11037 |
| [4] | Hernández, V., Solution IV of problem 10490, Amer. Math. Monthly, 106, 589-590 (1999) |
| [5] | Hoffman, M. E., Quasi-symmetric functions and mod \(p\) multiple harmonic sums (2007), preprint |
| [6] | Lehmer, D. H., Interesting series involving the central binomial coefficients, Amer. Math. Monthly, 92, 449-457 (1985) · Zbl 0645.05008 |
| [7] | Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0487.20007 |
| [8] | Staver, T. B., Om summasjon av potenser av binomialkoeffisienten, Norsk Mat. Tidsskrift, 29, 97-103 (1947) · Zbl 0030.28901 |
| [9] | Sun, Z. H., Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105, 193-223 (2000) · Zbl 0990.11008 |
| [10] | Sun, Z. W.; Tauraso, R., New congruences for central binomial coefficients, Adv. in Appl. Math., 45, 125-148 (2010) · Zbl 1231.11021 |
| [11] | Tauraso, R., New harmonic number identities with applications (2009), preprint |
| [12] | Van der Poorten, A., A proof that Euler missed … Apèry’s proof of the irrationality of \(\zeta(3)\), Math. Intelligencer, 1, 195-203 (1979) · Zbl 0409.10028 |
| [13] | Zhao, J., Bernoulli numbers, Wolstenholme’s theorem, and \(p^5\) variations of Lucas’ theorem, J. Number Theory, 123, 18-26 (2007) · Zbl 1115.11005 |
| [14] | Zhao, J., Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4, 73-106 (2008) · Zbl 1218.11005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
