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A fundamental modular identity and some applications

1993; American Mathematical Society; Volume: 61; Issue: 203 Linguagem: Inglês

10.1090/s0025-5718-1993-1197509-7

1088-6842

Richard Blecksmith, John Brillhart, Irving Gerst,

Finite Group Theory Research

We prove a six-parameter identity whose terms have the form x α T ( k 1 , l 1 ) T ( k 2 , l 2 ) {x^\alpha }T({k_1},{l_1})T({k_2},{l_2}) , where T ( k , l ) = ∑ − ∞ ∞ x k n 2 + l n 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 H(x)G({x^{11}}) - {x^2}G(x)H({x^{11}}) = 1 , where G ( x ) = ∏ n = 0 ∞ [ ( 1 − x 5 n + 1 ) ( 1 − x 5 n + 4 ) ] − 1 G(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})]}^{ - 1}}} and H ( x ) = ∏ n = 0 ∞ [ ( 1 − x 5 n + 2 ) ( 1 − x 5 n + 3 ) ] − 1 H(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 2}})(1 - {x^{5n + 3}})]}^{ - 1}}} . Two other identities, called "balanced Q 2 {Q^2} identities", are also established through its use.