June 10, 2011
Location: Kromme Nieuwegracht 80 (Utrecht), room 130
Complete monotonicity for inverse powers of some combinatorially defined polynomials
If P is a univariate or multivariate polynomial with real coefficients and strictly positive constant term, and β is a positive real number, it is sometimes of interest to know whether P−β has all nonnegative Taylor coefficients. For instance, Szegő showed in 1933 that for any n≥1, the polynomial
| n | ||||
| Pn(y1, …, yn) | = | ∑ | ∏ | (1-yj) |
| i=1 | j≠i |
has the property that Pn-β has nonnegative Taylor coefficients for all β≥1/2. But Szegő's proof was surprisingly indirect, exploiting Sonine-type integrals for triple products of Bessel functions. Here (in joint work with Alex Scott) I put Szegő's result in a much wider combinatorial and analytic context. We give elementary proofs of a vast generalization of Szegő's result, including a positive solution to a long-standing open problem of Lewy and Askey. More precisely, 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. The 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 — together with a variety of constructions that, given such polynomials, can create other ones with the same property. Our methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones).
Reference
A.D. Scott and A.D.S., Complete monotonicity for inverse powers of some combinatorially defined polynomials, in preparation.
The deformed exponential function F(x,y) = ∑n≥0 xn yn(n-1)/2 / n! and a plethora of related things
The "deformed exponential function" F(x,y) = ∑n≥0 xn yn(n-1)/2 / n! arises in the enumeration of connected graphs and, more generally, in the study of the Potts model on the complete graphs Kn. I here consider F as an analytic function of x,y defined for x∈C and |y|≤1, and I investigate its roots xk(y). I formulate a series of intriguing conjectures, almost all of which remain unproven (though some of them have been verified by power-series expansion through order y899). A few of these conjectures can, however, be proven when F is replaced by the "partial theta function" Θ0(x,y) = ∑n≥0 xn yn(n-1)/2, which arises in the theory of q-series and in particular in Ramanujan's "lost" notebook. I mention also some surprising connections with the theory of integrable systems in classical mechanics, notably the Calogero–Moser dynamics.
Presentation
The slides of this presentation are available here.
References
A.D.S., Some wonderful conjectures (but almost no theorems) at the boundary between analysis, combinatorics and probability: The entire function F(x,y), the polynomials PN(x,w), and the generating polynomials of connected graphs, available here.
A.D.S., The leading root of the partial theta function, arXiv:1106.1003.
A.D.S., Roots of a formal power series f(x,y) = ∑n≥0 an(y) xn, with applications to graph enumeration and q-series, Series of 4 lectures at Queen Mary (London), available here.
A.D.S., A ridiculously simple and explicit implicit function theorem, Seminaire Lotharingien de Combinatoire 61A (2009), article B61Ad, available here or at arXiv:0902.0069.