
A Generalization of the Weyl Algebra 307
from Se
s;h
(x)=
x
0
dt
(1−hst)
1
s
by a standard integration. Let us turn to the case s =0.
Using (8.52), one finds S
0;h
(n, 1) = h
n−1
S(n, 1) = h
n−1
and, consequently, Se
0;h
(x)=
n≥1
h
n−1
x
n
n!
=
1
h
(e
hx
− 1). In the case s = 1, we use in a similar fashion (8.56) and find
S
1;h
(n, 1) = (−h)
n−1
s(n, 1) = h
n−1
(n − 1)!, implying that
Se
1;h
(x)=
n≥1
h
n−1
(n − 1)!
x
n
n!
=
1
h
n≥1
(hx)
n
n
=
1
h
log
1
1 − hx
,
as asserted.
Example 8.80 Let h =1and s =2. It follows from Proposition 8.79 that
Se
2;1
(x)=1−
√
1 − 2x.
According to [1037, Example 5.2.6], this is the exponential generating function of binary
set bracketings such that if b(n) is the number of