Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions (Q342946)

Let \(a,b,\lambda, q, |q|<1\), are complex numbers. Define NEWLINE\[NEWLINE G(a,b,\lambda):=G(a,\lambda;b;q):=\sum_{n \geq 0} \frac{q^{(n^2+n)/2}(a+\lambda)\cdots(a+\lambda q^{n-1})}{(1-q)\cdots (1-q^n)(1+bq)\cdots(1+bq^n)}. NEWLINE\]NEWLINE Ramanujan defined the following continued fraction NEWLINE\[NEWLINE \frac{G(aq,b,\lambda q)}{G(a,b,\lambda)}=\frac{1}{1}+\frac{aq+\lambda q}{1}+\frac{bq+\lambda q^2}{1}+\frac{aq^2+\lambda q^3}{1} + \frac{bq^2+\lambda q^4}{1}+\dots NEWLINE\]NEWLINE It is well known that Rogers-Ramanujan continued fractions, Ramanujan cubic continued fractions and Göllnitz-Gordon continued fractions are special cases of Ramanujan continued fractions. The \(q\)-analog of Gauss's continued fractions is Heine's continued fractions. In this paper, the authors show that various continued fractions for the quotient of general Ramanujan functions \(G(aq,b,\lambda q)/G(a,b,\lambda)\) may be derived each other via Baue-Muir transformations. Also they show that these continued fractions may be derived from either Heine's continued fraction for ration of hypergeometric functions \(_2\phi_1\). A number of new versions of some continued fraction expansions of Ramanujan for certain combinations of infinite products are derived.