Joint Complete Monotonicity of reciprocal of a polynomial in two variables
Mandar Khasnis, V. M. Sholapurkar
TL;DR
The paper investigates when the reciprocal of a two-variable polynomial $p(x,y)=b(x)+a(x)y$ with $\deg a<\deg b$ yields a joint completely monotone net $\{\frac{1}{p(m,n)}\}_{m,n}$, focusing on the bi-degree $(k,1)$ with $l=k-1$. It proves a sufficient interlacing condition $b_1\le a_1\le b_2\le \cdots \le a_l\le b_k$ under which the net is joint CM, and a necessary inequality $\sum_{j=1}^{l}\frac{1}{a_j} \le \sum_{j=1}^{k}\frac{1}{b_j}$, while also providing non-CM examples in the regime $l<k-1$. The work connects these nets to CDSP and to Hilbert-space models, showing how $1/p(m,n)$ induces Hausdorff moment sequences and subnormal, essentially normal, weighted-shift operators on reproducing kernel Hilbert spaces. Through a decomposition of $\frac{b(x)}{a(x)}$ and a Hausdorff-moment analysis, the paper links algebraic root interlacing to operator-theoretic subnormality, yielding concrete criteria and illustrating limitations to generalization. Overall, it advances partial classification of joint CM nets arising from two-variable polynomials and provides a channel between moment problems and subnormal operator theory.
Abstract
In this article, we study some special cases of the problem of classifying polynomials $p:\mathbb{R}^2_+\to (0,\infty)$ for which the net $\{\frac{1}{p(m,n)}\}_{m,n\in \mathbb{Z}_+}$ is a completely monotone net, where $p(x,y)=b(x)+a(x)y$, $a(x)$ and $b(x)$ are polynomials with $deg(a) < deg (b)$. We also give examples of $a(x)$ and $b(x)$ such that the net $\{\frac{1}{p(m,n)}\}_{m,n\in \mathbb{Z}_+}$ is not completely monotone. Furthermore, we also study some properties of the associated subnormal weighted $2$-shifts.