Lisa Jacobsen

Domains of validity for some of Ramanujan's continued fraction formulas

Abstract By establishing uniform convergence of some of Ramanujans continued fractions, we are able to extend the domain of validity for some of his formulas. We also suggest more direct proofs for some of his results based on this technique.

Convergence acceleration of limit periodic continued fractions under asymptotic side conditions

SummaryWe present a method of convergence acceleration for limitk-periodic continued fractionsK(an/1) orK(1/bn) satisfying certain asymptotic side conditions. The method represents an improvement of the “fixed point modification” considered by Thron and Waadeland [8], under these conditions. The regularC-fraction expansions of hypergeometric functions2F1(a, 1;c; z) and2F1(a, b; c; z)/2F1(a, b+1;c+1;z) are examples of continued fractions satisfying these conditions.

Even and odd parts of limit periodic continued fractions

Abstract Even and odd parts of limit periodic continued fractions K( a n /1), a n → a are again limit periodic in most cases. For a = ∞ additional conditions are needed; the particular ones used in the paper imply that the even and odd parts K( c n /1) and K( d n /1) are such that c n → − 1 4 and d n → − 1 4 . Illustrating examples are included.

Repeated modifications of limitk-periodic continued fractions

SummaryThe advantages of using modified approximants for continued fractions, can be enhanced by repeating the modification process. IfK(an/bn) is limitk-periodic, a natural choice for the modifying factors is ak-periodic sequence of right or wrong tails of the correspondingk-periodic continued fraction, if it exists. If the modified approximants thus obtained are ordinary approximants of a new limitk-periodic continued fraction, we repeat the process, if possible. Some examples where this process is applied to obtain a convergence acceleration are also given.

On the convergence of limit periodic continued fractions K(a n/1), wherea n→−1/4. Part III

Previous results on the convergence and divergence of K(an/1).an→−1/4, are generalized by constructing a sequence of reference continued fractions having explicit tails and associated chain sequences and then applying Pincherles theorem together with a perturbation theory for solutions to the associated difference equations.

On the Bauer-Muir transformation for continued fractions and its applications

Abstract We show that the Bauer-Muir transformation is useful also for continued fractions with non-positive elements. Some examples of applications of this transformation are studied.

Remarks to a definition of convergence acceleration illustrated by means of continued fractions K( a n /1) where a n –> 0

Abstract Let {cn} and {c∗n} be two sequences of complex numbers converging to the finite limit c. We say that {c∗n} converges faster to c than {cn} if (c ∗ n −c) (c n −c) →0 . Convergence acceleration is based on this definition. In this paper we draw attention to some of its deficiencies. As an example we consider the case where {cn} is the sequence of approximants of a continued fraction K ( a n 1 ) where an → 0.

Meromorphic continuation of functions given by limit k -periodic continued fractions

Abstract We prove a method for meromorphic continuation of functions ƒ(z) given by limit k-periodic continued fractions K(an(z)⧸bn(z)) (pointwise limit) such that 1. (i) ƒ is analytic in some domain D o, 2. (ii) an, bn are entire functions, 3. (iii) a kn+p (z)→ a p (z) , b kn+p (z)→ b p (z) geometrically and fast enough in a domain D , D ∩ D 0 ≠ O; i.e., | a kn+p (z)− a p (z) | ⩽ C(z)q(z) n , | b kn+p (z)− b p (z) | ⩽ C(z)q(z) n for a q(z), q(z)

General correspondence for continued fractions

We introduce a new concept of correspondence for continued fractions, based on modified approximants.

THREE COMPUTATIONAL ASPECTS OF CONTINUED FRACTION/PADÉ APPROXIMANTS

After a general presentation of continued fractions and their connections to power series and Pade tables the following aspects are discussed, mainly through examples: 1. Speed of convergence and a priori truncation error estimates. 2. Acceleration of convergence. 3. Analytic continuation and numerical stability.