Convergence questions for limit periodic continued fractions

Abstract

Using one of the standard notations we then have wv è" +1 + b n+2 + + b N + w Instead of S$(w) we shall usually write S N (w).The continued fractionthen is the ordered pair «{#"}, {&"}>, {£ w (0)}>. Here it is understood that {a n } and {b n } be such that S n (0) is defined as an extended complex number for all n (or at least from a certain n = n 0 on). This is in particular the case if a n ^ 0 for all « or if a n = 0 for all n and simultaneously b n ^ 0. The sequences {a n } and {£"} are called the sequences of elements, and(5 M (0)} is the sequence of approximants. Convergence of a continued fraction means convergence of the sequence of approximants (possibly to oo ). In case of convergence the notation K™ =1 (ajb n ) is also used for lim^oo S;(0). The approximants of a continued fraction can also be represented as S,(0) = AJB n9 where

An Error Estimate for Continued Fractions

“…Recently obtained results in error analysis/acceleration by Thron and Waadeland [6,7] apply only in the case a ¥= 0, and the author's results along these lines [2] require the sort of bounds described in the present paper.…”

Converging factors for continued fractions 𝐾(𝑎_{𝑛}/1), 𝑎_{𝑛}→0

“…Waadeland, in essence, employed ft, = a in his study of limit-periodic T-fractions [7]. More recently, Thron and Waadeland [6] reported the far more general result that follows. _ Theorem 2.…”

Meromorphic Extension of Analytic Continued Fractions Across Their Divergence Line with Applications to Orthogonal Polynomials

“…In all these cases the divergence line S of (12) and, if (13) holds for some R > 1, the boundary curves S(R) of regions into which meromorphic extension of T(z) across S is possible are described explicitly. The method of meromorphic extension developed in the present paper generalizes and also simplifies a method used by the author in [9,11] in the special case of (1), where all b" = 0 and in the special case of (12) where all d" = 0.…”

“…Another method of meromorphic extension which is entirely different from the one used in the present paper is the general method of "modified" continued fractions developed and investigated in the work of Gill [3], Jacobsen [4,5], Thron and Waadeland [13,14].…”