Ordered product expansions of operators (AB)±m with arbitrary positive integer.

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 ℚ- and ℙ-ordered forms of (QP)^±m are also obtained by using an analogy method. [ABSTRACT FROM AUTHOR]