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Level 6: Ramanujan’s Cubic Continued Fraction

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Ramanujan's Theta Functions

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Abstract

We prove that

$$\displaystyle{ \frac{q^{1/3}} {1 + \frac{q + q^{2}} {1 + \frac{q^{2} + q^{4}} {1 + \frac{q^{3} + q^{6}} {1 + \cdots } }}} = q^{1/3}\prod _{ j=1}^{\infty }\frac{(1 - q^{6j-5})(1 - q^{6j-1})} {(1 - q^{6j-3})^{2}} }$$

and conduct an extensive study of the infinite product.

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Authors and Affiliations

  1. Institute of Natural and Mathematical Science, Massey University, Auckland, New Zealand

    Shaun Cooper

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  1. Shaun Cooper
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Cooper, S. (2017). Level 6: Ramanujan’s Cubic Continued Fraction. In: Ramanujan's Theta Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-56172-1_7

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