Two calculation formulas of expansion each other for normal ordering and antinormal ordering of creation operator and annihilation operator are given in this paper as () n( +) m=∑pk=0m!n!(m-k)!(n-k)!k!( +) m-k () n-k and ( +) n() m=∑pk=0(-1) km!n!(m-k)!(n-k)!k!() m-k ( +) n-k , where p=min(n,m). It is right that we prove by using mathematical inductive method. The two formulas of this paper are used in a wide class of study of quantum statistical properties for various excited optical fields.