Lisa Lorentzen

Continued fractions with applications

I. Introductory Examples. II. More Basics. III. Convergence Criteria. IV. Continued Fractions and Three-Term Recurrence Relations. V. Correspondence of Continued Fractions. VI. Hypergeometric Functions. VII. Moments and Orthogonality. VIII. Pade Approximants. IX. Some Applications in Number Theory. X. Zero-free Regions. XI. Digital Filters and Continued Fractions. XII. Applications to Some Differential Equations. Appendix: Some Continued Fraction Expansions. References. Index.

Compositions of contractions

We prove that if D is a simply connected (open) domain in the complex plane C, E is a closed subset of D, and {fn∞n=1} are functions analytic in D such that fn(D)⊆ E for all n, then the compositions Fn(z) = f1of2o⋯ofn(z) for n = 1, 2, 3,…, converge uniformly in D to a constant function. Some generalizations are also presented.

Computation of limit periodic continued fractions. A survey

Over the last 20 years a large number of algorithms has been published to improve the speed and domain of convergence of continued fractions. In this survey we show that these algorithms are strongly related. Actually, they essentially boil down to two main principles.We also prove some results on asymptotic expansions of tail values of limit periodic continued fractions.

Continued fractions with circular twin value sets

We prove that if the continued fraction K(a n /1) has circular twin value sets (V 0 , V 1 ), then K(a n /1) converges except in some very special cases. The results generalize previous work by Jones and Thron.

GENERAL CONVERGENCE IN QUASI-NORMAL FAMILIES

Montel introduced the concept of quasi-normal families

A convergence question inspired by Stieltjes and by value sets in continued fraction theory

f:\varOmega\to\mathbb{C}

Approximants of Sleszynski-Pringsheim continued fractions

in 1922:

Ideas from Continued Fraction Theory Extended to Padé Approximation and Generalized Iteration

\mathcal{F}

A note on separate convergence for continued fractions

is quasi-normal of order

Bestness of the parabola theorem for continued fractions

N