Domains of holomorphy of generating functions of Pólya frequency sequences of finite order (Q1826898)

The paper deals with the class of generating functions of Pólya frequency sequences of finite order \(r,r\in {\mathbb N},\) which is denoted by \(PF_r.\) (These sequences are also called \(r\)-multiply positive sequences by Fekete, Pólya, Schoenberg). The author studies domains of holomorphy of generating functions of such sequences. The main result of the paper is the following theorem: a domain \(G\) is the domain of holomorphy for a generating function of Pólya frequency sequence if and only if \(0 \in G, \) \(G\) is symmetric with respect to the real axis and \(\text{dist}(0,\partial G)=\text{dist}(0, \partial G\cap (0, \infty )).\)