Classical Partition Identities and Basic Hypergeometric Series
15
B1.3. Classical generating functions. When S = N, the set of natural
numbers, the corresponding generating functions may be displayed, respectively, as
∞
Y
1
1
=
(q; q)∞
1 − qm
m=1
=
∞
X
p(n) qn
(B1.3a)
n=0
∞
Y
X
1
1
=
=
p` (n) x` qn
m
(qx; q)∞
1
−
xq
m=1
(−q; q)∞ =
∞
Y
(−qx; q)∞ =
m=1
∞
Y
(1 + qm ) =
`,n≥0
∞
X
Q(n) qn
(B1.3b)
(B1.3c)
n=0
(1 + xqm ) =
m=1
X
Q` (n) x` qn .
(B1.3d)
`,n≥0
Manipulating the ge