Classical Partition Identities and Basic Hypergeometric Series
25
where the q-Gauss binomial coefficient is defined by
(q; q)m+n
m+n
=
.
m q
(q; q)m (q; q)n
Proof. For these two formulae, it is sufficient to prove only one identity
because
p` (n|m) = p` (n|m) − p`−1(n|m).
In fact, supposing that (B4.2b) is true, then (B4.2a) follows in this manner:
X
X
X
p` (n|m) qn =
p`(n|m) qn −
p`−1 (n|m) qn
n≥0
n≥0
n≥0
`+m−1
`+m
`−1+m
=
−
= q`
.
m−1 q
m q
m
q
Now we should prove (B4.2b). Extracting the coefficient of x` from the
generation function (B4.1b), we get
∞
X
p` (n|m) qn = [x`]
n=0
1
.
(x; q)m+1
Observing that the function 1/(x; q)m+1 is analytic at x = 0 for |q| < 1, we
can expand it into MacLaurin series:
∞
X
1
=
Bk (q)xk
(x; q)m+1
k=0
where the coefficients Bk (q) are independent of x to be determinated.
Reformulating it under replacement x → qx as
∞
X
1
=
Bk (q)xk qk
(qx; q)m+1
k=0
and then noting further that both fractions just displayed differ in factors
(1 − x) and (1 − xqm+1 ), we have accordingly the following:
(1 − x)
∞
X
Bk (q)xk = (1 − xqm+1 )
k=0
∞
X
Bk (q)xk qk .
k=0
`
Extracting the coefficient of x from both sides we get
B` (q) − B`−1 (q) = q` B` (q) − qm+` B`−1 (q)
which is equivalent to the following recurrence relation
B` (q) = B`−1 (q)
1 − qm+`
1 − q`
for ` = 1, 2, · · · .