- Title
- Some cubic modular identities of Ramanujan
- Creator
- Borwein, J. M.; Borwein, P. B.; Garvan, F. G.
- Relation
- Transactions of the American Mathematical Society Vol. 343, Issue 1, p. 35-47
- Publisher Link
- http://dx.doi.org/10.1090/S0002-9947-1994-1243610-6
- Publisher
- American Mathematical Society (AMS)
- Resource Type
- journal article
- Date
- 1994
- Description
- There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is ([formula cannot be replicated] qn2+nm+m2)³ = ([formula cannot be replicated] ωn-mqn²+nm+m²)³ + ([formula cannot be replicated] q(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²)³. Here ω = exp(2π i/3). In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.
- Subject
- theta functions; q-series; eta function; modular forms; cubic modular equations; hypergeometric functions
- Identifier
- http://hdl.handle.net/1959.13/940448
- Identifier
- uon:13009
- Identifier
- ISSN:0002-9497
- Rights
- First published in Transactions of the American Mathematical Society in Vol. 343, No. 1, pp. 35-47, 1994, published by the American Mathematical Society.
- Language
- eng
- Full Text
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