Multivariable versions of a lemma of Kaluza's

Abstract

Let $d\in \mathbb{N}$ and $f(z)= \sum_{\alpha\in \mathbb{N}_0^d} c_\alpha z^\alpha$ be a convergent multivariable power series in $z=(z_1,\dots,z_d)$. In this paper we present two conditions on the positive coefficients $c_\alpha$ which imply that $f(z)=\frac{1}{1-\sum_{\alpha\in \mathbb{N}_0^d} q_\alpha z^\alpha}$ for non-negative coefficients $q_\alpha$. If $d=1$, then both of our results reduce to a lemma of Kaluza's. For $d>1$ we present examples to show that our two conditions are independent of one another. It turns out that functions of the type $$f(z)= \int_{[0,1]^d} \frac{1}{1-\sum_{j=1}^d t_j z_j} d\mu(t)$$ satisfy one of our conditions, whenever $d\mu(t) = d\mu_1(t_1) \times \dots \times d\mu_d(t_d)$ is a product of probability measures $\mu_j$ on $[0,1]$. Our results have applications to the theory of Nevanlinna-Pick kernels.