  We introduce and study a deformation of commutative polynomial algebras in
even numbers of variables. We also discuss some connections and applications of
this deformation to the generalized Laguerre orthogonal polynomials and the
interchanges of right and left total symbols of differential operators of
polynomial algebras. Furthermore, a more conceptual re-formulation for the
image conjecture [Z3] is also given in terms of the deformed algebras.
Consequently, the well-known Jacobian conjecture [Ke] is reduced to an open
problem on this deformation of polynomial algebras.
