Lucas congruences for the Ap\'ery numbers modulo $p^2$

TL;DR

This paper investigates the Apéry numbers, providing a formula for their generating function's coefficients in terms of multiple zeta values, and establishes Lucas congruences modulo p^2 for certain integers based on their base-p digits.

Contribution

It introduces a new formula for the Taylor coefficients of the Apéry numbers' generating function and proves Lucas congruences modulo p^2 for specific integers.

Findings

Taylor coefficients expressed via multiple zeta values

Lucas congruences modulo p^2 for certain integers

Conditions on base-p digits for congruences

Abstract

The sequence $A(n)_{n \geq 0}$ of Ap\'ery numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.