Partitions related to positive definite binary quadratic forms. (English) Zbl 1457.11140
In this paper, three \(q\)-series identities are presented that are motivated by connections to positive definite quadratic forms.
Specifically, let \(P_{m,j}(n)\) be the number of partitions of \(n\) into \(m\) distinct parts, of which \(j\) is the largest, and one part \(2m+1\) that may appear any number of times (or not at all). Then the following identity holds:
\[ \sum_{m,n,j \geq 0} (-1)^j P_{m,j}(n)q^n = \frac12 \sum_{n \geq 0} q^{n^2+n/2} (1+q^{n+1/2}) \sum_{|j| \leq n} q^{j^2/2} + \frac12 \sum_{n \geq 0} (-1)^n q^{n^2+n/2} (1-q^{n+1/2}) \sum_{|j| \leq n} (-1)^j q^{j^2/2}. \]
The other two identities are similar, but involve overpartition pairs rather than ordinary partitions.
Specifically, let \(P_{m,j}(n)\) be the number of partitions of \(n\) into \(m\) distinct parts, of which \(j\) is the largest, and one part \(2m+1\) that may appear any number of times (or not at all). Then the following identity holds:
\[ \sum_{m,n,j \geq 0} (-1)^j P_{m,j}(n)q^n = \frac12 \sum_{n \geq 0} q^{n^2+n/2} (1+q^{n+1/2}) \sum_{|j| \leq n} q^{j^2/2} + \frac12 \sum_{n \geq 0} (-1)^n q^{n^2+n/2} (1-q^{n+1/2}) \sum_{|j| \leq n} (-1)^j q^{j^2/2}. \]
The other two identities are similar, but involve overpartition pairs rather than ordinary partitions.
Reviewer: Stephan Wagner (Uppsala)
MSC:
| 11P81 | Elementary theory of partitions |
| 05A15 | Exact enumeration problems, generating functions |
| 11E16 | General binary quadratic forms |
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