1 research outputs found
Ramanujan and Extensions and Contractions of Continued Fractions
If a continued fraction is known to converge
but its limit is not easy to determine, it may be easier to use an extension of
to find the limit. By an extension of
we mean a continued fraction whose odd or even part is . 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 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.Comment: 16 page