Document Type

Article

Publication Date

2007

Abstract

If a continued fraction K∞n=1an/bn is known to converge but its limit is not easy to determine, it may be easier to use an extension of K∞n=1an/bn to find the limit. By an extension of K∞n=1an/bn we mean a continued fraction K∞n=1cn/dn whose odd or even part is K∞n=1an/bn. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if K∞n=1an/bn is the even part); (ii) Find the limit of the other contraction and show that the odd and even parts of the extension tend to the same limit. We apply these ideas to derive new proofs of certain continued fraction identities of Ramanujan and to prove a generalization of an identity involving the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith and Brillhart.

Publication Title

The Ramanujan Journal

ISSN

1382-4090

Publisher

Springer

Volume

14

Issue

3

First Page

389

Last Page

404

Comments

Preprint version is available here.

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