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The continued fraction in the title is perhaps the deepest of Ramanujan’s q-continued fractions. We give a new proof of this continued fraction, more elementary and shorter than the only known proof by Andrews, Berndt, Jacobsen, and Lamphere. On page 45 in his lost notebook, Ramanujan states an asymptotic formula for a continued fraction generalizing that in the title. The second main goal of this paper is to prove this asymptotic formula.

On Ramanujan’s continued fraction for (𝑞²;𝑞³)_{∞}/(𝑞;𝑞³)_{∞}