Using the coherent state representation we derive some new operator identities and study some mathematical relations in combinatorics. The technique of integral within an ordered product (IWOP) of operators plays an essential role in realizing our goal.
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.