
66 Commutation Relations, Normal Ordering, and Stirling Numbers
From the many properties of the partial Bell polynomials (see [280, 935]), we recall the
recurrence relation derived by Bell himself [74, Equation (4.2)],
B
n+1
(y
1
,...,y
n+1
)=
n
k=0
n
k
B
n−k
(y
1
,...,y
n−k
)y
k+1
. (3.28)
If y
k
= x for all k ≥ 1, then this reduces due to (3.27) to Theorem 3.29(5). Furthermore, if
y = y(x) denotes a function of x, and if the derivatives of y with respect to x are denoted
by y
= dy/dx = Dy, y
= Dy
and y
(k)
= Dy
(k−1)
,thene
−y
D
n
e
y
= B
n
(y, y
,y
,...,y
(n)
);
see [935, Page 35]. Before closing this section, let us mention the recent paper [981], where
some of the above consider ...