Based on the explicit Weyl-ordered form of Wigner operator and the technique of integration within Weyl-ordered product of operators we derive the Weyl-ordered operator product formula. The formula is then generalized to the entangled form with the help of entangled state representations.
We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. By virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Weyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Weyl correspondence. The Wigner function's zero-point value is also discussed.