A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables

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.