Employing the general ordering theorem, operational methods and the incomplete 2-dimensional Hermite polynomials we have derived the t-ordered expansion of the Fock space projectors. Using the result, a new integration formula regarding incomplete 2-dimensional Hermite polynomials is obtained.
By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.
The structure of ordered expansions in powers of boson operators and of canonical operators and the dual problem of operator reconstruction from ordered moments is derived and applied to the complete Gaussian class of ordering. In particular, the interpolation lines between normal and antinormal ordering and between standard and antistandard ordering with Weyl symmetrical ordering in their centre are dealt with in detail. The auxiliary operators for expansions in symmetrical ordering are explicitly found in the Fock-state representation and in other different representations. General and specialized formulae are derived for different ordering of powers of linear combinations of boson and of canonical operators which involve Hermite polynomials of operators. The link between symmetrical ordering of powers of boson operators and of canonical operators is expressed by means of Jacobi polynomials. Some basic formulae of operator ordering and operator expansion are collected for convenient use in the appendix.