Some supercongruences occurring in truncated hypergeometric series

By Ling Long, Ravi Ramakrishna. 2014

158 citations recorded by the source corpus.

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 $\Gamma$-values. At this point sums have become quotients. Writing these $\Gamma$-quotients as $\Gamma_p$-quotients, we are in a situation that is well-suited for proving $p$-adic congruences. These $\Gamma_p$-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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5f45b33b7b7c28a5303f1f094e2e20ac8eae65fe

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Citations in the corpus

On Van Hamme's (A.2) and (H.2) supercongruences (2018, methodology): “Finally, we combine the solutions to represent mySum as follows: In[4] := FindLinearCombination[recSol,mySum, 2] Out[4] = 3 + 4n 4(1 + 2n)2 n ∏ ι1=1 4ι21 (−1 + 2ι1) 2 This implies that both sides of (3.5) are equal.”
Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions (2015, methodology): “which was shown by the third author and Ramakrishna in [27], while numerically we see that”
On the (K.2) supercongruence of Van Hamme (2015, unknown): “Primary 11B65; Secondary 33C20, 33F10.”
Hypergeometric Functions over Finite Fields (2015, methodology): “See [20] for a general result, and [63] by the second and third authors, [85] by the fourth author, [71] by Osburn and Zudilin for some recent developments on Ramanujan-type supercongruences.”
On the supercongruence conjectures of van Hamme (2015, methodology): “eometric series identities and evaluations. Long also uses a similar but more general method in [9] to prove (J.2). Furthermore, (D.2) has now been proved by Long and Ramakrishna in a recent preprint [10]. In addition, they prove that (H.2) holds modulo p3 when p ≡ 1 (mod 4), and provide extensions ”