Congruences for Apéry numbers βn =∑k=0nn k2n+k k

In this paper, we establish some congruences involving the Apéry numbers [Formula: see text]. For example, we show that [Formula: see text] for any positive integer [Formula: see text], and [Formula: see text] for any prime [Formula: see text], where [Formula: see text] is the [Formula: see text]th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Aṕery numbers, [Formula: see text] and [Formula: see text].

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the [Formula: see text]-Lucas numbers.

Hankel-type determinants for some combinatorial sequences

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer [Formula: see text], define [Formula: see text] [Formula: see text] For [Formula: see text], we show that [Formula: see text] and [Formula: see text] are positive odd integers, and [Formula: see text] and [Formula: see text] are always integers.

Arithmetic properties of Apéry-like numbers

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$p$.