16
CHU Wenchang and DI CLAUDIO Leontina
Proof. For S = N, the generating function of {p` (n|N)} reads as
X
X X
p` (n) x`qn =
x`
p` (n)qn
`,n≥0
=
`≥0
∞
Y
k=1
n≥0
1
1
=
.
k
1 − xq
(qx; q)∞
m
Extracting the coefficient of x , we get
∞
X
pm (n) qn = [xm ]
n=0
1
.
(qx; q)∞
For |q| < 1, the function 1/(qx; q)∞ is analytic at x = 0. We can therefore
expand it in MacLaurin series:
∞
X
1
=
A` (q)x`
(B2.2)
(qx; q)∞
`=0
where the coefficients A` (q) are independent of x to be determined. Performing the replacement x → x/q, we can restate the expansion just
displayed as
∞
X
1
=
A` (q)x` q−` .
(B2.3)
(x; q)∞
`=0
It is evident that (B2.2) equals (1 − x) times (B2.3), which results in the
functional equation
∞
X
A` (q)x` = (1 − x)
`=0
∞
X
A` (q)x` q−` .
`=0
Extracting the coefficient of x
m
from both expansions, we get
Am (q) = Am (q)q−m − Am−1 (q)q1−m
which is equivalent to the following recurrence relation
q
Am (q) =
Am−1 (q) where m = 1, 2, · · · .
1 − qm
Iterating this recursion for m-times, we find that
Am (q) =
(1 −
qm A0 (q)
qm
A0 (q).
=
m−1
−q
) · · · (1 − q)
(q; q)m
qm )(1
Noting that A0 (q) = 1, we get finally
∞
X
n=0
pm (n) qn = [xm]
1
qm
=
.
(qx; q)∞
(q; q)m
This completes the proof of Proposition B2.1.