A $q$-continued fraction (1901.00584v1)

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.