Hirschman-Widder densities

TL;DR

This paper explores the properties of Hirschman-Widder densities, a class of Pólya frequency functions, analyzing their behavior under polynomial transformations and their connections to moments and Schur polynomials.

Contribution

It characterizes when polynomial transformations of Hirschman-Widder densities remain Pólya frequency functions and links their moments to Schur polynomial representations.

Findings

Polynomial of a Hirschman-Widder density is a Pólya frequency only if it is a homothety.

Certain densities have all positive-integer powers as Pólya frequency functions.

Connections established between moments, Maclaurin coefficients, and density recovery.

Abstract

Hirschman and Widder introduced a class of P\'olya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a P\'olya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a P\'olya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.