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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A fundamental modular identity and some applications


Authors: Richard Blecksmith, John Brillhart and Irving Gerst
Journal: Math. Comp. 61 (1993), 83-95
MSC: Primary 11P83; Secondary 05A19, 11F11
DOI: https://doi.org/10.1090/S0025-5718-1993-1197509-7
MathSciNet review: 1197509
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Abstract: We prove a six-parameter identity whose terms have the form ${x^\alpha }T({k_1},{l_1})T({k_2},{l_2})$, where $T(k,l) = \sum \nolimits _{ - \infty }^\infty {{x^{k{n^2} + l n}}}$. This identity is then used to give a new proof of the familiar Ramanujan identity $H(x)G({x^{11}}) - {x^2}G(x)H({x^{11}}) = 1$, where $G(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})]}^{ - 1}}}$ and $H(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 2}})(1 - {x^{5n + 3}})]}^{ - 1}}}$. Two other identities, called "balanced ${Q^2}$ identities", are also established through its use.


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Keywords: Triple and quintuple product, modular identity, balanced <IMG WIDTH="29" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${T^2}$"> and <IMG WIDTH="31" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="${Q^2}$"> identity
Article copyright: © Copyright 1993 American Mathematical Society