Some supercongruences occurring in truncated hypergeometric series

@article{Long2014SomeSO,
  title={Some supercongruences occurring in truncated hypergeometric series},
  author={Ling Long and Ravi Ramakrishna},
  journal={arXiv: Number Theory},
  year={2014}
}

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

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

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

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

<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

Supercongruences for rigid hypergeometric Calabi–Yau threefolds

Some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial

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

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

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

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

  • Victor J. W. GuoLong Li
  • Mathematics
    Revista 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

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

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

(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

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

Lectures on p-adic Differential Equations

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

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

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