The class of generating functions for completely monotone sequences (moments of finite positive measures on
[
0
,
1
]
[0,1]
) has an elegant characterization as the class of Pick functions analytic and positive on
(
−
∞
,
1
)
(-\infty ,1)
. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on
[
0
,
1
]
[0,1]
. Also we provide a simple analytic proof that for any real
p
p
and
r
r
with
p
>
0
p>0
, the Fuss-Catalan or Raney numbers
r
p
n
+
r
(
p
n
+
r
n
)
\frac {r}{pn+r}\binom {pn+r}{n}
,
n
=
0
,
1
,
…
n=0,1,\ldots
, are the moments of a probability distribution on some interval
[
0
,
τ
]
[0,\tau ]
if and only if
p
≥
1
p\ge 1
and
p
≥
r
≥
0
p\ge r\ge 0
. The same statement holds for the binomial coefficients
(
p
n
+
r
−
1
n
)
\binom {pn+r-1}n
,
n
=
0
,
1
,
…
n=0,1,\ldots \,
.