## 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…