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arXiv · 1901.00584

A $q$-continued fraction

Douglas Bowman·James Mc Laughlin·Nancy J. Wyshinski
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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.

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Douglas Bowman, James Mc Laughlin, Nancy J. Wyshinski. 2019-01-03. A $q$-continued fraction. https://doi.org/10.1142/s179304210600070x

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