On the modified convergence of some continued fractions of Rogers- Ramanujan type
type: Item
instance of: scholarly article
title: On the modified convergence of some continued fractions of Rogers- Ramanujan type
zbMATH Open document ID: 0813.40001
DOI: 10.1016/0097-3165(94)90021-3
author: Krishnaswami Alladi
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-05-21T00:00:00Z
review text: In the analytic theory of continued fractions, the method of modified convergence is known for some time. This boils down to replacing the tail of a continued fraction by a fixed or variable number different from zero (that is what happens when a continued fraction is just `cut off'). The author uses for modifier the replacement of the tail starting with \(a_ n\) by \(a_ n + b_{n+1}\), where \(b_{n+1}\) is the next partial numerator. He then proves modified convergence for three different types of \(q\)-continued fractions which can be seen as companions to well known identities of Rogers-Ramanujan and of Gordon. The modified limits are different from the ordinary continued fraction limit in two of the cases and in one of the cases the continued fraction has different limits for the sequences of even and odd approximants and therefore is divergent in the ordinary sense. But the modified continued fraction does converge and the limit is just the mediant of the `odd' and `even' limits.
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 554398
zbMATH Keywords: continued fractions of Rogers-Ramanujan type
zbMATH Keywords: method of modified convergence
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90021-3
OpenAlex ID: W1997665818
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description: scientific article; zbMATH DE number 554398