13 research outputs found
Ramanujan and the Regular Continued Fraction Expansion of Real Numbers
In some recent papers, the authors considered regular continued fractions of
the form where ,
and are integers. The limits of such continued fractions, for
general and in the cases and , were given as ratios of certain
infinite series.
However, these formulae can be derived from known facts about two continued
fractions of Ramanujan. Motivated by these observations, we give alternative
proofs of the results of the previous authors for the cases and and
also use known results about other -continued fractions investigated by
Ramanujan to derive the limits of other infinite families of regular continued
fractions.Comment: 14 page
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
REAL NUMBERS WITH POLYNOMIAL CONTINUED FRACTION EXPANSIONS
Abstract. In this paper we show how to apply various techniques and theorems (including Pincherle’s theorem, an extension of Euler’s formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like π and e, ζ(k) (for each positive integer k ≥ 2), various special functions evaluated at integral arguments and various algebraic numbers. We also pose several questions about the nature of the set of real numbers which have a polynomial continued fraction expansion. 1