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 2φ1 functions, or other similar continued fraction expansions of ratios of 2φ1 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)q 1 − abq2 − (a − bq2)(b − aq2)q 1 − abq4 − (1 − a2q2)(1 − b2q2)q3 1 − abq6 − (a − bq4)(b − aq4)q3 1 − abq8 − ··· .
Journal of Mathematical Analysis and Applications
Lee, J., McLaughlin, J., & Sohn, J. (2017). Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions. Journal of Mathematical Analysis and Applications, 447(2), 1126-1141. http://dx.doi.org/doi.org/10.1016/j.jmaa.2016.10.052