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

In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq,b,λq)/G(a,b,λ) 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 either Heine's continued fraction for a ratio of ϕ12 functions, or other similar continued fraction expansions of ratios of ϕ12 functions. Further, by employing essentially the same methods, a new continued fraction for G(aq,b,λq)/G(a,b,λ) is derived. 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:(−a,b;q)∞−(a,−b;q)∞(−a,b;q)∞+(a,−b;q)∞=(a−b)1−ab−(1−a2)(1−b2)q1−abq2−(a−bq2)(b−aq2)q1−abq4−(1−a2q2)(1−b2q2)q31−abq6−(a−bq4)(b−aq4)q31−abq8−⋯.