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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.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1197509-7
Keywords: Triple and quintuple product, modular identity, balanced $ {T^2}$ and $ {Q^2}$ identity
Article copyright: © Copyright 1993 American Mathematical Society

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