## Complete monotonicity for inverse powers of some combinatorially defined polynomials    [PDF]

Alexander D. Scott, Alan D. Sokal
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 Szego 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^{-\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^{-\beta} for some \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.
View original: http://arxiv.org/abs/1301.2449