Complete monotonicity for inverse powers of some combinatorially defined polynomials

Abstract

We prove the complete monotonicity on (0, ∞) 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 −β 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 −β for some β > 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.

Positivity of rational functions and their diagonals

“…, which is just a rescaled version of 1/e 2 (1 − x, 1 − y, 1 − z, 1 − w). As already mentioned in the introduction, the non-negativity of this rational function was recently established in [23]. The scaled initial diagonal terms s n := 9 n [(xyzw) n ]h 2/3,0,0 (x, y, z, w) are 1, 24, 1080, 58560, 3490200, 220739904, .…”

“…The question to decide whether a given rational function is positive, that is, whether its Taylor coefficients are all positive, goes back to Szegő [25] and has since been investigated by many authors including Askey and Gasper [3,4,5], Koornwinder [17], Ismail and Tamhankar [14], Gillis, Reznick and Zeilberger [11], Kauers [15], Straub [24], Kauers and Zeilberger [16], Scott and Sokal [23]. The interested reader will find a nice historical account in [23]. A particularly interesting instance is the Askey-Gasper rational function A(x, y, z) :…”

“…also referred to as the Lewy-Askey problem. Very recently, Scott and Sokal [23] succeeded in proving the non-negativity of (2), both in an elementary way by an explicit Laplace-transform formula and based on more general results on the basis generating polynomials of certain classes of matroids. Note that by a result from [16] the positivity of (2) would follow from the positivity of D(x, y, z, w) :…”

Exponential varieties

“…The following recent result by Scott and Sokal [34] motivates studying hyperbolic polynomials in the context of exponential families. It shows that any homogeneous polynomial that is (up to a negative power) the partition function of an exponential family must be hyperbolic.…”

“…These are associated with hyperbolic polynomials and their hyperbolicity cones [18,26]. Work of Scott and Sokal [34] implies that every exponential family whose canonical parameters form a convex cone and whose partition function is the power of a homogeneous polynomial must be hyperbolic (Theorem 3.3). We conjecture that the converse holds as well, namely that one can build a (statistical) exponential family, i.e.…”