On the Generalized Rogers–Ramanujan Continued Fraction

On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are examined. One of them is an identity bearing a superficial resemblance to the generating function for the generalized Rogers–Ramanujan continued fraction. Thus, our third main goal is to establish, with the help of an idea of F. Franklin, a partition bijection to prove this identity.