  This article investigates second order intertwinings between semigroups of
birth-death processes and discrete gradients on the space of natural integers
N. It goes one step beyond a recent work of Chafa{\"i} and Joulin which
establishes and applies to the analysis of birth-death semigroups a first order
intertwining. Similarly to the first order relation, the second order
intertwining involves birth-death and Feynman-Kac semigroups and weighted
gradients on N, and can be seen as a second derivative relation. As our main
application, we provide new quantitative bounds on the Stein factors of
discrete distributions. To illustrate the relevance of this approach, we also
derive approximation results for the mixture of Poisson and geometric laws.
