4 results
Proof of some conjectural congruences involving Apéry and Apéry-like numbers
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 07 March 2024, pp. 508-527
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In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then
\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}where
$A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.
ON THE DIVISIBILITY OF SUMS INVOLVING APÉRY-LIKE POLYNOMIALS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 106 / Issue 2 / October 2022
- Published online by Cambridge University Press:
- 14 March 2022, pp. 203-208
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- October 2022
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We prove a divisibility result on sums involving the Apéry-like polynomials
$$ \begin{align*} V_n(x)=\sum_{k=0}^n {n\choose k}{n+k\choose k}{x\choose k}{x+k\choose k}, \end{align*} $$
which confirms a conjectural congruence of Z.-H. Sun. Our proof relies on some combinatorial identities and transformation formulae.
CONGRUENCES FOR THE
$(p-1)$TH APÉRY NUMBER
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 28 November 2018, pp. 362-368
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- June 2019
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We prove two conjectural congruences on the
$(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.
Arithmetic properties of Apéry-like numbers
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 2 / February 2018
- Published online by Cambridge University Press:
- 20 October 2017, pp. 249-274
- Print publication:
- February 2018
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We provide lower bounds for
$p$-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the 
$p$-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the 
$p$-Lucas property for almost all primes 
$p$.
