# Stefan Richter: Multivariable versions Kaluza's Lemma

**Time: **
Fri 2022-11-04 10.00 - 11.00

**Location: **
Albano house 1, floor 3, Cramérrummet

**Participating: **
Stefan Richter (University of Tennessee)

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.