Ramanujan-type supercongruences.(English)Zbl 1231.11147

The starting point of this paper comes from the observation that several of Ramanujan’s and Ramanujan–like formulas for $$1/\pi$$ admit natural conjectural $$p$$-adic analogues. The thesis set forward in this paper is that perhaps all known Ramanujan’s formulas for $$1/\pi$$ and even the ones for $$1/\pi^2$$ admit conjectural $$p$$-adic congruences. The author presents several such conjectural congruences and proves three of them. The proofs use the WZ–method of creative telescoping together with some classical $$p$$-adic congruences such as Wolstenholme’s theorem. The paper concludes with an experimental section presenting several conjectural cogruences modulo $$p^3,~p^5$$ or higher powers of $$p$$, of which the author can prove only a few usually modulo a smaller power of $$p$$ than experiments seem to indicate. Due to the abundance of conjectural supercongruences it is likely that this paper will spur a lot of interest in the area of $$p$$-adic Ramanujan supercongruences.

MSC:

 11Y55 Calculation of integer sequences 33C20 Generalized hypergeometric series, $${}_pF_q$$ 11B65 Binomial coefficients; factorials; $$q$$-identities 11F33 Congruences for modular and $$p$$-adic modular forms

Zbl 1207.33012
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References:

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