  We prove that of the creator operators, on the $d$ commuting indeterminates
polynomial algebra, are linearly independent. We further study the connection
between the classical (one dimensional) and the multi-dimensional
($d$-dimensional, $d \geq 1$) Favard Lemmas. Moreover, we investigate the
dependence of the Jacobi sequences on the linear change of basis of
$\mathbb{C}^d$. Finally we prove that the Jacobi sequences associated to the
probability measure product on $\mathbb{R}^d$ are diagonal matrices in the
basis introduced by the tensor product of the orthogonal polynomials of the
factor measures.
