Ramanujan-type supercongruences

Abstract

We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/{\pi }^2$, and prove three of these examples.

10 pages

References in corpus (1)

• A p-adic analogue of a formula of Ramanujan

Cited by in corpus (46)

• A $q$-microscope for supercongruences
• Open Conjectures on Congruences
• Dwork-type supercongruences through a creative $q$-microscope
• On the (K.2) supercongruence of Van Hamme
• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• On a $q$-deformation of modular forms
• Supercongruences for sporadic sequences
• A common $q$-analogue of two supercongruences
• Some new $q$-congruences for truncated basic hypergeometric series
• A $p$-adic supercongruence for truncated hypergeometric series ${}_7{F}_6$
• Ramanujan-type formulae for $1/\pi$: $q$-analogues
• A supercongruence for generalized Domb numbers
• Symbolic summation methods and hypergeometric supercongruences
• $p$-adic analogues of hypergeometric identities
• A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
• Some supercongruences occurring in truncated hypergeometric series
• Lost in translation
• Mosaic supercongruences of Ramanujan-type
• Products and sums divisible by central binomial coefficients
• Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions
• Proof of two supercongruences conjectured by Z.-W. Sun
• Ramanujan-Sato-like series
• The method of creative microscoping
• Kind of proofs of Ramanujan-like series
• Proof of two supercongruences by the Wilf-Zeilberger method
• Bilateral sums related to Ramanujan-like series
• Apéry's theorem and problems for the values of Riemann's zeta function and their $q$-analogues
• Supercongruences for truncated hypergeometric series and p-adic gamma function
• Proof of some supercongruences via the Wilf-Zeilberger method
• Some congruences related to a congruence of Van Hamme
• ($q$-)Supercongruences hit again
• On supercongruences for truncated sums of squares of basic hypergeometric series
• On two congruences involving Franel numbers
• Factors of some truncated basic hypergeometric series
• Polynomial Reduction and Super Congruences
• q-analogues of the G.2 Supercongruence of Van Hamme
• Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm
• Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method
• $N$-tuple sum analogues for Ramanujan-type congruences
• Streamlined WZ method proofs of Van Hamme supercongruences
• On van Hamme's (A.2) and (H.2) supercongruences
• Factors of certain sums involving central q-binomial coefficients
• Some q-supercongruences from Rahman's summation formula
• Some variations of a "divergent" Ramanujan-type $q$-supercongruence
• A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$
• Supercongruences for sums involving Domb numbers