Convergent Star product on $\mathbb C^n$ and Star Functions

Star product is an associative product for polynomials on $\mathbb C^n$ and is regarded as a one parameter deformation of the usual commutative product.
We extend the product to certain class of entire functions on $\mathbb C^n$. We discuss the convergence of products and the intertwinners. Due to the convergence, we define a star function for entire function by replacing, in expansion, its each monomial of the usual product with the one given by star product. We show some concrete examples of star functions and discuss certain identities.