A new proof of a conjecture of Hirschhorn and Sellers on overpartitions
A new proof of a conjecture of Hirschhorn and Sellers on overpartitions
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DOI：
10.1007/s11139-014-9598-z
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发表时间：
2014-08
期刊：
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通讯作者：
Bernard L. S. Lin
中科院分区：
文献类型：
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作者：
Bernard L. S. Lin
记为的超分割数。赫什霍恩和塞勒斯对此进行了解释。最近，Chen和Xia利用Ramanujan，赫什霍恩和Sellers给出的θ函数的2-剖分公式以及Alaca，Alaca和威廉姆斯给出的θ函数的-参数化证明了这个猜想.本文利用Ramanujan的一些恒等式给出了这个猜想的一个新的证明。我们还证明了Choi，Kang和洛夫乔伊关于曲柄的奇偶性加权的划分的Ramanujan型恒等式与Ramanujan的“最美恒等式”等价，其中表示的划分数.
Letdenote the number of overpartitions of. Hirschhorn and Sellers conjectured that for,. Recently, Chen and Xia have proven this conjecture using 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, as well as-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we shall give a new proof of this conjecture using some identities of Ramanujan. We also show that Ramanujan type identity for partitions weighted by the parity of the crank due to Choi, Kang and Lovejoy is equivalent with Ramanujan’s “Most Beautiful Identity” on, wheredenotes the number of partitions of.