Some properties of associated spaces with sub-Hankel determinants

TL;DR

This paper investigates the structure of spaces linked to sub-Hankel determinants, revealing their properties as non-reductive prehomogeneous vector spaces, and explores their Legendre transforms and related b-functions.

Contribution

It characterizes the associated space as a non-reductive, regular prehomogeneous vector space and derives formulas for sub-Hankel determinants with orthogonal polynomial components.

Findings

Identified the space as non-reductive and prehomogeneous

Computed multiplicative Legendre transforms of sub-Hankel determinants

Established relations between b-functions of polarization and sub-Hankel determinants

Abstract

In this note, we show that the space associated with sub-Hankel determinant is a non-reductive, regular prehomogeneous vector space, and we give the multiplicative Legendre transforms of sub-Hankel determinants. Moreover we observe certain relations between $b$-functions of polarization of PVpolynomials and $b$-functions of sub-Hankel determinants, and give some formulas about sub-Hankel determinants whose components are orthogonal ponlynomials.