Classical Partition Identities and Basic Hypergeometric Series 19
From the first generating function , we see that the bivariate generating function of partitions into parts ≤ k reads as
∞∑
l , n = 0 p l ( n | k ) x l q n =
1 ( qx ; q ) k
.
Putting an extra part λ 1 = k with enumerator xq k over the partitions enumerated by the last generating function , we therefore derive the bivariate generating function of partitions into l parts with the first one λ 1 = k as follows :
∞∑ p l ( n | λ 1 = k ) x l q n = xqk .
( qx ; q ) k l , n = 0
Classifying the partitions of n into exactly l parts with each parts ≤ m according to the first part λ 1 = k , we get the following expression
∞∑
l , n = 0 p l ( n | m ) x l q n = m∑
∞∑
k = 0 l , n = 0
= 1 + x m∑
k = 1
p l ( n | λ 1 = k ) x l q n
q k ( qx ; q ) k which is the second identity .
□
B3 . Partitions into distinct parts and the Euler formula
B3.1 . Theorem . Let Q m ( n ) be the number of partitions into exactly m distinct parts . Its generating function reads as
∞∑ Q m ( n ) q n = n = 0 q ( 1 + m
2 ) ( 1 − q )( 1 − q 2 ) ···( 1 − q m ) . ( B3 . 1 )
Proof . Let λ =( λ 1 > λ 2 ··· > λ m > 0 ) be a partition enumerated by Q m ( n ). Based on λ , define another partition µ =(µ 1 ≥ µ 2 ≥ ··· ≥ µ m ≥ 0 ) by µ k := λ k − ( m − k + 1 ) for k = 1, 2 , ···, m . ( B3.2 )