
48 Commutation Relations, Normal Ordering, and Stirling Numbers
where
f denotes the compositional inverse of f ,that,isf (f(t)) = f(f(t)) = t.Observethat
(2.26) is of the form (2.22). The associated raising and lowering operators are then given by
M
g,f
=
X −
g
(D)
g(D)
1
f
(D)
,P
f
= f (D), (2.27)
see [940, Theorem 3.7.1]. The order of terms in M
g,f
is important; note that X enters
only linearly. M
g,f
is called Sheffer shift. In this situation, we say that u
n
(x) is Sheffer for
(g(t),f(t)) and we also denote this by u
n
(x) ∼ (g(t),f(t)). Clearly, x
n
∼ (1,t).
Example 2.79 Let P (D)=
1
2
D, that is, f (t)=
1
2
t.Letg(t)=e
t
2
/4
. Then we find from
(2.27) that M(X, D)=2X − D