Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions

Abstract

In this paper we show that various continued fractions for the quotient of general Ramanujan functions $G(aq,b,łq)/G(a,b,ł)$ may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from Heine's continued fraction for a ratio of $_2ϕ_1$ functions and other continued fractions of a similar type, and by this method derive a new continued fraction for $G(aq,b,łq)/G(a,b,ł)$. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: \begin{multline*} \frac{(-a,b;q)_{\infty} - (a,-b;q)_{\infty}}{(-a,b;q)_{\infty}+ (a,-b;q)_{\infty}} = \frac{(a-b)}{1-a b} \- \frac{(1-a^2)(1-b^2)q}{1-a b q^2}\\ \- \frac{(a-bq^2)(b-aq^2)q}{1-a b q^4} %\phantom{sdsadadsaasdda}\\ \- \frac{(1-a^2q^2)(1-b^2q^2)q^3}{1-a b q^6} \- \frac{(a-bq^4)(b-aq^4)q^3}{1-a b q^8} \- \cds . \end{multline*}