New approach for anti-normally and normally ordering bosonic-operator functions in quantum optics

In this paper, we provide a new kind of operator formula for anti-normally and normally ordering bosonic-operator functions in quantum optics, which can help us arrange a bosonic-operator function f(lambda Q + vP) in its anti-normal and normal ordering conveniently. Furthermore, mutual transformation formulas between anti-normal ordering and normal ordering, which have good universality, are derived too. Based on these operator formulas, some new differential relations and some useful mathematical integral formulas are easily derived without really performing these integrations.