Classical Partition Identities and Basic Hypergeometric Series 23
over 0 ≤ m ≤∞, we find the following identity: ∞∏
( 1 + xq 1 + 2n) = n = 0
∞∑
m = 0 x m q m2( q 2; q 2) m
where the left hand side is the bivariate generating function of the partitions into odd distinct parts.
It is trivial to verify that under replacements x →−xq −1 / 2 and q → q 1 / 2 the last formula is exactly the identity displayed in( B3.3).
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Unfortunately, there does not exist the closed form for the generating function of Q m( n), numbers of partitions into ≤ m distinct parts.
B3.3. Dually, if we classify the partitions into distinct parts ≤ m according to their maximum part. Then we can derive the following finite and infinite series identities
m∏ m∑ k−1
∏( 1 + q j x) = 1 + x q k( 1 + q i x)( B3.4a)
j = 1
j = 1 k = 1
∞∏ ∞∑
( 1 + q j x) = 1 + x k = 1 i = 1
∏
( 1 + q i x).( B3.4b)
q k k−1 i = 1
Proof. For the partitions into distinct parts with the maximum part equal to k, their bivariate generating function is given by
k−1
∏ q k x( 1 + q i x) which reduces to 1 for k = 0.
i = 1
Classifying the partitions into distinct parts ≤ m according to their maximum part k with 0 ≤ k ≤ m, weget m∑( −qx; q) m = 1 + x q k( −qx; q) k−1. k = 1
The second identity follows from the first one with m →∞.
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