The results obtained seem reasonable, although I have no time to verify all the derivations.
A second example is the equation 9b for Wigner function: the associated quotation is ref. 17 by one of the authors and M.A. Kasevich in Phys Rev A (2006). A brief search on the web proves that a similar equation was used by Hongyi Fan Phys Rev A 656, 064102 (2002) who quotes two references dating from 1984 and a book published in 1994. I am not sufficiently aware of the use f Wigner function but I am sure that there are better and more useful references to quote in addition to ref 17!