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

A Deformation of Commutative Polynomial Algebras in Even Numbers of Variables