Non-formal star product and star functions

Star product is considered as a generalization of the Moyal product, which is a one parameter deformation of ususal multiplication of functions and is associative. The star products are also parametrized by matrices giving non-commutative or commutative algebraic structures on function space according to the matrices.
In this talk, we will deal with non-formal star products on complex space $\mathbb{C}^{n}$. For polynomials on $\mathbb{C}^{n}$, replacing usual multiplication by that of star product gives star polynomials, and similarly for power series of entire functions on $\mathbb{C}^{n}$ we obtain star functions. The star funtions of entire function have singularities in general.
We give a review on the star products and star functions, and discuss some problems. We consider mainly concrete examples of star functions.