Generalizations of Ramanujan's reciprocity theorem and their applications
Generalizations of Ramanujan's reciprocity theorem and their applications
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DOI：
10.1112/jlms/jdl002
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发表时间：
2007-02
期刊：
影响因子：
--
通讯作者：
Soon-Yi Kang
中科院分区：
文献类型：
--
作者：
Soon-Yi Kang
首先，我们简要地回顾了与部分theta函数相关的q-级数的Ramanujan倒易定理，并给出了该定理的一个新证明。接下来，我们推导出了互易定理的推广，它也是1 × 1求和公式和Jacobi三重积恒等式的推广，并证明了这些互易定理也导致了五重积恒等式的推广。最后，我们给出了广义互易定理和乘积恒等式的一些应用，包括六个平方和的生成函数和超划分的生成函数的新表示。
First, we briefly survey Ramanujan's reciprocity theorem for a certain q‐series related to partial theta functions and give a new proof of the theorem. Next, we derive generalizations of the reciprocity theorem that are also generalizations of the 1ψ1 summation formula and Jacobi triple product identity and show that these reciprocity theorems lead to generalizations of the quintuple product identity, as well. Last, we present some applications of the generalized reciprocity theorems and product identities, including new representations for generating functions for sums of six squares and those for overpartitions.