A Theorem on T-Fractions Corresponding to a Rational Function

A convergence theorem for limitärperiodisch 𝑇-fractions of rational functions

“…A step in the proof is to establish the following lemma (see [6, p. Remark 2. The existence of an uncountable set of rational functions with nontrivial limitärperiodisch £-fractions is proved in [1]. Applying the functions used in this proof we can prove the existence of (an uncountable set of) rational functions with poles in |z|<l and with nontrivial limitärperiodisch £-fractions.…”

“…where /^_1), ßk0) are arbitrary (complex) constants, and let {/"} and {dn} be the sequences defined in (2) and 3 where the constants ßkn) are given by certain recursion formulas (see [1]).…”

The classical irrationality problem for $T$-fractions

“…This problem is mentioned in the classic book [5, p. 174] and recent contributions to this problem area by Jefferson, Hag and Waadeland, are reported and commented in [4, p. 380]. See also [1][2][3]7]. In the next paragraph we give a solution of this problem in the form of necessary and sufficient conditions involving a polynomial sequence satisfying certain properties.…”

Singularities of a class of meromorphic functions

“…This enables one to establish fairly general convergence criteria for F-fractions. Recent work on F-fractions includes results of Waadeland [16], [17], [18] on convergence of F-fraction expansions of certain functions holomorphic in circular discs, as well as articles by Jones and Thron [5], Jefferson [4] and Hag [3]. Jones and Thron give, among other results, some theorems on the location of singular points of functions represented by F-fractions all of whose elements dn are positive.…”