Level 8: The Ramanujan–Göllnitz–Gordon Continued Fraction

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

Chapter Chapter 8 in Ramanujan's Theta Functions, 2017, pp 467-507 from Springer

Abstract: Abstract We prove that q 1 ∕ 2 1 + q + q 2 1 + q 3 + q 4 1 + q 5 + q 6 1 + q 7 + ⋯ = q 1 ∕ 2 ∏ j = 1 ∞ ( 1 − q 8 j − 7 ) ( 1 − q 8 j − 1 ) ( 1 − q 8 j − 5 ) ( 1 − q 8 j − 3 ) $$\displaystyle{ \frac{q^{1/2}} {1 + q + \frac{q^{2}} {1 + q^{3} + \frac{q^{4}} {1 + q^{5} + \frac{q^{6}} {1 + q^{7} + \cdots }}}} = q^{1/2}\prod _{ j=1}^{\infty }\frac{(1 - q^{8j-7})(1 - q^{8j-1})} {(1 - q^{8j-5})(1 - q^{8j-3})}}$$ and obtain results that are analogues of theorems in Chapters 5, 6, and 7.

Date: 2017
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DOI: 10.1007/978-3-319-56172-1_9

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