
198 Commutation Relations, Normal Ordering, and Stirling Numbers
Now, we would like to recall some normal ordering results discussed already in Sec-
tion 5.3. The result (6.45) is due to McCoy [789] and Schwinger [985] (but see also the
closely related work of Sylvester [1051]), while (6.47) seems to be due to Mehta [795, 796]
and Wilcox [1142].
Theorem 6.30 Let U and V be generators of the extended Weyl algebra
ˆ
A
1
satisfying
[U, V ]=1. Then one has for any λ ∈ C
e
λV U
=:e
(e
λ
−1)VU
: . (6.45)
If N = VU, then we can write (6.45) equivalently as
e
λN
=:B(λ, N):, (6.46)
where B(x, y)=e
(e
x
−1)y
denotes the exponential generating function of the Bell polynomi-
als. ...