A natural derivative on [0,n] and a binomial Poincaré inequality

We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator del(n), defined as a linear combination of left and right finite differences. We show that this operator del(n) plays a key role in a new Poincare (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.