CHAPTER B
Generating Functions of Partitions
For a complex sequence {αn|n = 0, 1, 2, · · ·}, its generating function with a
complex variable q is defined by
A(q) :=
∞
X
αn q n
αn = [qn] A(q).
n=0
When the sequence has finite non-zero terms, the generating function reduces to a polynomial. Otherwise, it becomes an infinite series. In that
case, we suppose in general |q| < 1 from now on.
B1. Basic generating functions of partitions
Given three complex indeterminates x, q and n with |q| < 1, the shifted
factorial is defined by
(x; q)∞
=
∞
Y
(1 − xqk )
k=0
(x; q)n
=
(x; q)∞
.
(qn x; q)∞
When n is a natural number in particular, it reduces to
(x; q)0 = 1
and (x; q)n =
n−1
Y
(1 − qk x) for n = 1, 2, · · · .
k=0
We shall frequently use the following abbreviated notation for shifted factorial fraction:
(a; q)n(b; q)n · · · (c; q)n
a, b, · · · , c
=
.
q
α, β, · · · , γ
(α; q)n(β; q)n · · · (γ; q)n
n