  The influence of the use of the generalized Hermite polynomial on the
Hermite-based lattice Boltzmann (LB) construction approach, lattice sets, the
thermal weights, moments and the equilibrium distribution function (EDF) are
addressed. A new moment system is proposed. The theoretical possibility to
obtain a high-order Hermite-based LB model capable to exactly match some first
hydrodynamic moments thermally 1) on-Cartesian lattice, 2) with thermal weights
in the EDF, 3) whilst the highest possible hydrodynamic moments that are
exactly matched are obtained with the shortest on-Cartesian lattice sets with
some fixed real-valued temperatures, is also analyzed.
  Keywords: Lattice Boltzmann, fluid dynamics, kinetic theory, distribution
function
