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Generalizations of Ramanujan's reciprocity theorem and their applications
Author(s) -
Kang SoonYi
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdl002
Subject(s) - ramanujan's sum , reciprocity (cultural anthropology) , mathematics , ramanujan theta function , reciprocity law , identity (music) , triple product , product (mathematics) , pure mathematics , algebra over a field , discrete mathematics , psychology , social psychology , physics , geometry , acoustics
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 ψ 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.