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Journal of the London Mathematical Society Advance Access originally published online on January 8, 2007
Journal of the London Mathematical Society 2007 75(1):18-34; doi:10.1112/jlms/jdl002
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© 2007 London Mathematical Society

Generalizations of Ramanujan's reciprocity theorem and their applications

Soon-Yi Kang

School of Mathematics
Korea Institute for Advanced Study
207-43 Cheongnyangni 2-dong
Dongdaemun-gu
Seoul 130-722
Korea
sykang{at}kias.re.kr

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 1{psi}1 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.

2000 Mathematics Subject Classification 33D15, 11B65, 05A15.

Received January 10, 2006; published online January 8, 2007.


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