Some properties of associated spaces with sub-Hankel determinants

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.