Operators' s-parameterized ordering and its classical correspondence in quantum optics theory

In reference to the Weyl ordering x(m)p(n) -> (1/2)(m) Sigma(m)(l=0) (m l) Xm-l P-n X-l, where X and P are coordinate and momentum operator, respectively, this paper examines operators; s-parametrized ordering and its classical correspondence, finds the fundamental function-operator correspondence (1 - s/2)((n+m)/2) H-m,H-n (root 2/1 - s alpha*, root 2/1 - s alpha) -> a(dagger m)a(n), and its complementary relation alpha(n)alpha*(m) -> (-i)(n+m) (1 - s/2)((m+n)/2) : H-m,H-n (i root 2/1 - s a(dagger), i root 2/1 - s a):, where H-m,H-n is the two-variable Hermite polynomial a, a(dagger) are bosonic annihilation and creation operators respectively, s is a complex parameter. The s'-ordered operator power-series expansion of s-ordered operator (s)a(dagger m)a(n)(s) in terms of the two-variable Hermite polynomial is also derived. Application of operators' s-ordering formula in studying displaced-squeezed chaotic filed is discussed.