Total nonnegativity of infinite Hurwitz matrices of entire and meromorphic functions

TL;DR

This paper characterizes entire and meromorphic functions that produce totally nonnegative infinite Hurwitz matrices, extending classical results and providing new criteria based on Stieltjes continued fractions.

Contribution

It generalizes the total nonnegativity characterization of Hurwitz matrices for real stable polynomials to broader classes of functions and introduces an alternative criterion for Pólya frequency sequences.

Findings

Complete description of functions generating totally nonnegative Hurwitz matrices

Extension of Asner and Kemperman's results to broader function classes

New criterion for entire functions to generate Pólya frequency sequences

Abstract

In this paper we fully describe functions generating the infinite totally nonnegative Hurwitz matrices. In particular, we generalize the well-known result by Asner and Kemperman on the total nonnegativity of the Hurwitz matrices of real stable polynomials. An alternative criterion for entire functions to generate a P\'olya frequency sequence is also obtained. The results are based on a connection between a factorization of totally nonnegative matrices of the Hurwitz type and the expansion of Stieltjes meromorphic functions into Stieltjes continued fractions (regular $C$-fractions with positive coefficients).