Complete monotonicity for inverse powers of some combinatorially defined polynomials

Abstract

We prove the complete monotonicity on ${(0,\infty )}^n$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that $P^{-\mathit{\beta }}$ is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of $P^{-\mathit{\beta }}$ for some $\mathit{\beta }>0$ can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P, and is also related to the Rayleigh property for matroids.