Joint Complete Monotonicity of reciprocal of a polynomial in two variables

TL;DR

This paper investigates conditions under which the reciprocal of certain bivariate polynomials forms a completely monotone net, and explores related properties of subnormal weighted 2-shifts, contributing to the understanding of polynomial monotonicity.

Contribution

It classifies specific polynomials in two variables for which the reciprocal forms a completely monotone net and examines related operator properties.

Findings

Identified conditions for complete monotonicity of reciprocal polynomials.

Provided examples where the reciprocal net is not completely monotone.

Analyzed properties of associated subnormal weighted 2-shifts.

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.