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 E Franklin, a partition bijection to prove this identity.
Key wordsRogers-Ramanujan continued fraction generalized Rogers-Ramanujan continued fraction Franklin’s involution Ramanujan’s lost notebook
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