A $q$-continued fraction

Abstract

We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for $(q^2;q^3{)}_{\infty }/(q;q^3{)}_{\infty }$ and $(q;q^2{)}_{\infty }/(q^3;q^6{)}_{\infty }^3$. In addition, we give a new proof of the famous Rogers-Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

23 pages

Cited by in corpus (2)

• Continued Fraction Proofs of $m$-versions of Some Identities of Rogers-Ramanujan-Slater Type
• Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions