26
CHU Wenchang and DI CLAUDIO Leontina
Iterating this relation `-times, we find that
B` (q) = B0 (q)
(qm+1 ; q)`
=
(q; q)`
m+`
`
q
where B0 (q) = 1 follows from setting x = 0 in the generating function
∞
X
1
=
Bk (q)xk .
(xq; q)m+1
k=0
Therefore we conclude the proof.
B4.3. Theorem. Classifying the partitions according to the number of
parts, we derive immediately two q-binomial identities (finite and infinite):
`+m
m
`=0
∞
X
`+m
x`
m
n
X
q`
=
=
`=0
m+n+1
n
m
Y
k=0
1
.
1 − xqk
(B4.3a)
(B4.3b)
Proof. In view of (B4.2a) and (B4.2b), the univariate generating functions for the partitions into parts ≤ m + 1 with the lengths equal to ` and
h
i
`+m
≤ n are respectively given by the q-binomial coefficients q`
and
m
h
i
m+n+1
. Classifying the partitions enumerated by the latter accorn
ding to the number of parts ` with 0 ≤ ` ≤ n, we establish the first identity.
By means of (B4.1b), we have
m
Y
k=0
∞
∞
∞
X
X
X
1
`
`
n
` m+`
=
x
p
(n|m)
q
=
x
`
1 − xqk
n=0
`=0
which is the second q-binomial identity.
`=0