
260 Commutation Relations, Normal Ordering, and Stirling Numbers
Note that by letting k = n − j, (7.73) transforms into a form similar to (7.70),
V
m
U
n
=
n
j=0
m + j − 1
j
n!
(n − j)!
h
j
U
n−j
V
m+j
.
Remark 7.45 Let us point out some history of (7.70) and (7.73). Both formulas have been
rediscovered independently several times. The starting point was the result of Benaoum (see
Proposition 7.42) in particular in its q-deformed version (see Proposition 7.60). Soon after
the publication of [77, 78] Zhang and Wang [1184] derived q-deformed variants of (7.70)
and (7.73). Even earlier, Rosengren [945, Remark 3.5] had derived a q-deformed variant of
(7.70) by specializin ...