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30
A q-microscope for supercongruences.pdf
A q-microscope for supercongruences
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36
Some supercongruences occurring in truncated hypergeometric series.pdf
Some supercongruences occurring in truncated hypergeometric series
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27
201403.5232v2 Some supercongruences occurring in truncated hypergeometric series.pdf
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p-adic Gamma functions that are stronger than those one can expect to prove using commutative formal group laws. We prove a number of such supercongruences by using classical hypergeometric transformation formulae. These formulae (see the appendix), most of which are decades or centuries old, allow us to write the terminating series as the ratio of products of of Γ-values. At this point sums have become quotients. Writing these Γ-quotients as Γp-quotients, we are in a situation that is well-suited for proving p-adic congruences. These Γp-functions can be p-adically approximated by their Taylor series expansions. Sometimes there is cancelation of the lower order terms, leading to stronger congruences. Using this technique we prove, among other things, a conjecture of Kibelbek and a strengthened version of a conjecture of van Hamme.
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16
some q-analogues of supercongruences of rodriguez-villegas:罗德里格兹的supercongruences这样一些:.pdf
NumberTheory 145 (2014) 301–316Contents lists available /locate/jntSome
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23
SUPERCONGRUENCES INVOLVING PRODUCTS OF ….pdf
SUPERCONGRUENCES INVOLVING PRODUCTS OF …
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19
MULTIVARIATE AP ERY NUMBERS AND SUPERCONGRUENCES OF RATIONAL….pdf
MULTIVARIATE AP ERY NUMBERS AND SUPERCONGRUENCES OF RATIONAL…
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14
201004.4337v4 ´Divergent´ Ramanujan-type supercongruences.pdf
"Divergent" Ramanujan-type series provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique
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8
Supercongruences for Apery-like numbers.pdf
Supercongruences for Apéry-like numbers
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19
201210.4489v3 Supercongruences and Complex Multiplication.pdf
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r=3, they are related to K3 surfaces. For special values of \lambda, with s=1 and r=3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the \lambda=1 case and confirm some other supercongruence conjectures at special values of \lambda.
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16
Supercongruences motivated by e.pdf
Supercongruences motivated by e
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