On Shifting Semicircular Roots

TL;DR

This paper investigates how shifting semicircular laws in continued fraction expansions of Stieltjes transforms affects the associated measures, providing explicit descriptions for shifts up to step two.

Contribution

It introduces a method to describe shifted semicircular laws via polynomial denominators in the measure's density, with explicit examples for low-level shifts.

Findings

Explicit formulas for shifted semicircular laws up to step two.

Demonstration of how polynomial denominators influence the measure.

Insights into the structure of shifted laws in continued fraction frameworks.

Abstract

In the framework of continued fraction expansions of Stieltjes transforms, we consider shifting of semicircular laws. The continuous part of the associated measure admits a density function which is the quotient of semicircular one by a polynomial. We study how this polynomial denominator determines shifted semicircular laws, with explicit descriptions in examples of shifting up to the level of step two.