Level 6: Ramanujan’s Cubic Continued Fraction

Shaun Cooper
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Shaun Cooper: Massey University, Institute of Natural and Mathematical Science

Chapter Chapter 6 in Ramanujan's Theta Functions, 2017, pp 361-422 from Springer

Abstract: Abstract We prove that q 1 ∕ 3 1 + q + q 2 1 + q 2 + q 4 1 + q 3 + q 6 1 + ⋯ = q 1 ∕ 3 ∏ j = 1 ∞ ( 1 − q 6 j − 5 ) ( 1 − q 6 j − 1 ) ( 1 − q 6 j − 3 ) 2 $$\displaystyle{ \frac{q^{1/3}} {1 + \frac{q + q^{2}} {1 + \frac{q^{2} + q^{4}} {1 + \frac{q^{3} + q^{6}} {1 + \cdots } }}} = q^{1/3}\prod _{ j=1}^{\infty }\frac{(1 - q^{6j-5})(1 - q^{6j-1})} {(1 - q^{6j-3})^{2}} }$$ and conduct an extensive study of the infinite product.

Date: 2017
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DOI: 10.1007/978-3-319-56172-1_7

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