  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.
