Backward Touchard congruence

Abstract

The celebrated Touchard congruence states that $B_{n+p}\equiv B_n+B_{n+1}$ (mod $p$), where $p$ is a prime number and $B_n$ denotes the $n$-th Bell number. In this paper we study divisibility properties of $B_{n-p}$ and their generalizations involving higher powers of $p$ as well as the $r$-Bell numbers. In particular, we show a close relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving \mbox{a new} relation between $r$-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo $p$.