Multivariable versions of a lemma of Kaluza's

TL;DR

This paper extends Kaluza's lemma to multivariable power series, providing conditions for representing functions as sums with non-negative coefficients, with applications to Nevanlinna-Pick kernels.

Contribution

It introduces two new independent conditions for multivariable power series to have non-negative coefficient representations, generalizing Kaluza's lemma.

Findings

Conditions are independent for dimensions greater than one.

Functions defined via integrals over product measures satisfy one of the conditions.

Results apply to the theory of Nevanlinna-Pick kernels.

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.