Lucas congruences for the Apery numbers modulo $p^2$

The sequence $A(n)_{n geq 0}$ of Apery 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 $p^2$.