Connecting the q-multiplicative convolution and the finite difference convolution

Abstract

In [1], Brändén, Krasikov, and Shapiro study root location preservation properties of finite difference operators. They then conjecture a characterization of linear finite difference operators on polynomials of fixed degree which preserve root mesh properties, via a natural polynomial convolution. We prove this conjecture using two methods. The first develops a novel connection between the additive (Walsh) and multiplicative (Grace-Szegö) convolutions, which can be generically used to transfer results from multiplicative to additive. We then use this to transfer an analogous result, due to Lamprecht [4], which demonstrates logarithmic root mesh preservation properties of a certain q-multiplicative convolution operator. The second method proves the result directly using a modification of Lamprecht's proof of the logarithmic root mesh result. We present his original argument in a streamlined fashion and then make the appropriate alterations to apply it to the additive case.