Ordered product expansions of operators (AB)±m with arbitrary positive integer**Project supported by the National Natural Science Foundation of China (Grant No. 11804085) and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2017MEM012).

Abstract

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.