We arrange quantum mechanical operators (a?a)m in their normally ordered product forms by using Touchard polynomials. Moreover, we derive the anti-normally ordered forms of (a?a)± m by using special functions as well as Stirling-like numbers together with the general mutual transformation rule between normal and anti-normal orderings of operators. Further, the ?- and ?-ordered forms of (QP)±m are also obtained by using an analogy method.