Joint Complete Monotonicity of reciprocal of a polynomial in two variables

In this article, we study some special cases of the problem of classifying polynomials $p:{\mathbb{R}}_{+}^2\to (0,\mathrm{\infty })$ 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.