Let $$d\in \mathbb N \text{ and } f(z)= \sum_{\alpha\in \mathbb{N}_0^d} c_\alpha z^\alpha$$ be a convergent multivariable power series in $$z=(z_1,\dots,z_d)$$. We present two independent 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 $q_\alpha$. 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 $$\mu_j \text{ on }[0,1]$$. The results have applications in the theory of Nevanlinna-Pick kernels.   This is joint work with Jesse Sautel.