The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials

TL;DR

This paper derives explicit formulas for the magnitude of odd-dimensional Euclidean balls using Hankel determinants of reverse Bessel polynomials, revealing their asymptotic behavior and providing combinatorial insights.

Contribution

It introduces a novel explicit formula for the magnitude of odd-dimensional balls in terms of Hankel determinants of reverse Bessel polynomials, expanding the understanding of magnitude invariants.

Findings

Magnitude is a rational function of the radius.

Explicit formulas involve Hankel determinants and combinatorial structures.

Asymptotic behavior of magnitude as radius grows is established.

Abstract

Magnitude is an invariant of metric spaces with origins in category theory. Using potential theoretic methods, Barcel\'o and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schr\"oder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.