18
CHU Wenchang and DI CLAUDIO Leontina
The corresponding generating function results in
∞
X
pm (n) qn =
n=0
∞
m X
X
pk (n) qn =
k=0 n=0
m
X
k=0
qk
.
(q; q)k
Recalling the first generating function expression (B2.4), we get the second
formula from the last relation.
B2.3. Gauss’ classical partition identity.
∞
Y
∞
X
1
xm
=
1
+
.
1 − xqn
(1 − q)(1 − q2) · · · (1 − qm )
n=0
m=1
(B2.5)
Proof. In fact, we have already established this identity from the demonstration of the last theorem, where it has been displayed explicitly in (B2.3).
Alternatively, classifying all the partitions with respect to the number of
parts, we can manipulate the bivariate generating function
1
(xq; q)∞
=
∞
X
=
`,n=0
∞
X
`=0
` n
p` (n)x q
=
∞
X
`=0
x
`
∞
X
p` (n)qn
n=0
x` q `
(1 − q)(1 − q2 ) · · · (1 − q` )
which is equivalent to Gauss’ classical partition identity.
B2.4. Theorem. Let p` (n|m) be the number of partitions of n with
exactly `-parts ≤ m. Then we have its generating function
∞
X
`,n=0
p`(n|m) x` qn =
1
.
(1 − qx)(1 − q2 x) · · · (1 − qm x)
The classification with respect to the maximum part k of partitions produces
another identity
X
1
qk
= 1+x
.
2
m
(1 − qx)(1 − q x) · · · (1 − q x)
(1 − qx)(1 − q2 x) · · · (1 − qk x)
m
k=1
Proof. The first generating function follows from (B1.1b).