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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