108 Citations
On some hypergeometric supercongruence conjectures of Long
- MathematicsThe Ramanujan Journal
- 2023
In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of…
Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series
- Mathematics
- 2018
Several new q -supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative…
Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series
- Materials ScienceConstructive Approximation
- 2020
Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative…
On the supercongruence conjectures of van Hamme
- Mathematics
- 2015
In 1997, van Hamme developed $$p$$p–adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs…
New $ q $-supercongruences arising from a summation of basic hypergeometric series
- MathematicsAIMS Mathematics
- 2021
<abstract><p>With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime…
Some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial
- Mathematics
- 2021
Abstract. In this paper, we establish some q-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of several summation and transformation formulas for basic…
A Whipple $$_7F_6$$ Formula Revisited
- MathematicsLa Matematica
- 2022
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$,…
Supercongruences for truncated hypergeometric series and p-adic gamma function
- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2018
Abstract We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by…
Some Supercongruences for Truncated Hypergeometric Series
- Mathematics2017 MATRIX Annals
- 2019
We prove various supercongruences involving truncated hypergeometric sums. These include a strengthened version of a conjecture of van Hamme. Our method is to employ various hypergeometric…
q-Supercongruences from squares of basic hypergeometric series
- MathematicsRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
- 2022
We give some new q-supercongruences on truncated forms of squares of basic hypergeometric series. Most of them are modulo the cube of a cyclotomic polynomial, and two of them are modulo the fourth…
References
SHOWING 1-10 OF 39 REFERENCES
A Gaussian hypergeometric series evaluation and Apéry number congruences
- Mathematics
- 2000
If p is prime, then let φp denote the Legendre symbol modulo p and let p be the trivial character modulo p. As usual, let n+1Fn(x)p := n+1Fn „ φp, φp, . . . , φp p, . . . , p | x « p be the Gaussian…
Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
- Mathematics
- 1993
We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric…
An extension of the Apéry number supercongruence
- Mathematics
- 2006
(modp). This was proved for primes p such that p ∤ a(p) by Ishikawa [8] and unconditionally by Ahlgren and Ono [1]. The a(p) for odd primes p are also known to be related to the modular Calabi–Yau…
Gaussian hypergeometric series and combinatorial congruences
- Mathematics
- 2001
We study the Gaussian hypergeometric series of type 3 F 2 over finite fields F p . For each prime p and each λ ∈ F p , we explicitly determine p 2 3 F 2(λ) p (mod p 2). Using this perspective, we are…
Supercongruences satisfied by coefficients of 2F1 hypergeometric series
- Mathematics
- 2009
Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular…
Lectures on p-adic Differential Equations
- Mathematics
- 1982
The present work treats p-adic properties of solutions of the hypergeometric differential equation d2 d ( x(l - x) dx + (c(l - x) + (c - 1 - a - b)x) dx - ab)y = 0, 2 with a, b, c in 4) n Zp, by…
Super congruences and Euler numbers
- Mathematics
- 2010
AbstractLet p > 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including…
A p-adic analogue of a formula of Ramanujan
- Mathematics
- 2007
Abstract.During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured p-adic…