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.
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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)