# Ramanujan and Cranks

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## Abstract

The existence of the crank was first conjectured by F. J. Dyson in 1944 and was later established by G. E. Andrews and F. C. Garvan in 1987. However, much earlier, in his lost notebook, Ramanujan studied the generating function *F* _{ a }(*q*) for the crank and offered several elegant claims about it, although it seems unlikely that he was familiar with all the combinatorial implications of the crank. In particular, Ramanujan found several congruences for *F* _{ a }(*q*) in the ring of formal power series in the two variables *a* and *q*. An obscure identity found on page 59 of the lost notebook leads to uniform proofs of these congruences. He also studied divisibility properties for the coefficients of *F* _{ a }(*q*) as a power series in *q*. In particular, he provided ten lists of coefficients which he evidently thought exhausted these divisibility properties. None of the conjectures implied by Ramanujan's tables have been proved.

## Keywords

Power Series Theta Function Formal Power Series Power Series Expansion Primitive Root## Preview

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