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

Abstract

The sequence A(n) n≥0 of Apéry numbers can be interpolated to C by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a 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 .