Mathematics – Number Theory
Scientific paper
2004-02-27
Mathematics
Number Theory
16 pages
Scientific paper
If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of $K_{n=1}^{\infty} a_{n}/b_{n}$ we mean a continued fraction $K_{n=1}^{\infty} c_{n}/d_{n}$ whose odd or even part is $K_{n=1}^{\infty} a_{n}/b_{n}$. 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 $K_{n=1}^{\infty}a_{n}/b_{n}$ 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.
Laughlin James Mc
Wyshinski Nancy J.
No associations
LandOfFree
Ramanujan and Extensions and Contractions of Continued Fractions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Ramanujan and Extensions and Contractions of Continued Fractions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ramanujan and Extensions and Contractions of Continued Fractions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-294494