Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.
We explore the theoretical possibility of extending the usual squeezed state to those produced by nonlinear singlemode squeezing operators. We derive the wave functions of exp[-（ig/2）（（1-X2）1/2P ＋ P（1-X2）1/2）]｜0 in the coordinate representation. A new operator＇s disentangling formula is derived as a by-product.
For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.
Based on the operator Hermite polynomials method（OHPM）, we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.
By virtue of the Weyl ordering method,we find a new formalism of optical field operator expansion in number state representation.Miscellaneous optical fields＇（coherent state,squeezed field,Wigner operator,etc.） new expansions are therefore exhibited.Some new generating functions of special polynomials are derived herewith.
By virtue of the operator Hermite polynomial method and the technique of integration within the ordered product of operators we derive a new kind of special function, which is closely related to one- and two-variable Hermite polynomials.Its application in deriving the normalization for some quantum optical states is presented.
By virtue of the density operator＇s P-representation in the coherent state representation, we derive a new quantum mechanical photon counting distribution formula. As its application, we calculate photon counting distributions for some given light fields. It is found that the pure squeezed state＇s photon counting distribution is related to the Legendre function, which is a new result.
