A formula for the convergents of a continued fraction of Ramanujan
Gaurav Bhatnagar, Michael D. Hirschhorn
TL;DR
This paper derives a new formula for the convergents of a generalized continued fraction related to Ramanujan's work, extending Ramanujan's original formula for the Rogers-Ramanujan continued fraction.
Contribution
It provides a novel explicit formula for the convergents of a broader class of continued fractions inspired by Ramanujan's entries.
Findings
Derived a formula for the convergents of a generalized continued fraction
Extended Ramanujan's original formula to a more general case
Enhances understanding of continued fractions in Ramanujan's notebooks
Abstract
In Entry 16, Chapter 16 of his notebooks, Ramanujan himself gave a formula for the convergents of the famous Rogers-Ramanujan continued fraction. We provide a similar formula for the convergents of a more general continued fraction, namely Entry 15 of Chapter 16.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics