  We consider probability mass functions $V$ supported on the positive integers
using arguments introduced by Caputo, Dai Pra and Posta, based on a
Bakry--\'{E}mery condition for a Markov birth and death operator with invariant
measure $V$. Under this condition, we prove a modified logarithmic Sobolev
inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux,
and Caputo, Dai Pra and Posta. We show how this inequality implies results
including concentration of measure and hypercontractivity, and discuss how it
may extend to higher dimensions.
