Analytic continuation of functions given by continued fractions, revisited
The author continues her investigations on meromorphic continuation of functions given by continued fractions \(K^ \infty_{n=0} a_ n(z)/b_ n(z)\). Let \(D\) be a domain where the continued fraction converges. The function \(f\) to which it converges may very well be meromorphically extendable to a larger domain \(D^*_ 0\), even if the continued fraction itself diverges in \(D^*_ 0 \backslash D\). The value of \(f(z)\) at a point in this set may, by a known idea, be obtained by using modified approximants. By introducing the convenient concepts of ``Ref'', characterizing reference continued fractions, ``Near'', characterizing continued fractions near a given one in a certain sense, and ``Uniconv'' for certain families of sequences of functions, the author establishes a general framework for the type of problems dealt with in this and earlier papers (by the author and others). The Ref- fractions will serve as a tool in the process of meromorphic continuation, and essential in their definition is that all elements be holomorphic in a given domain \(D^*\), that the continued fraction converges in a non-empty domain \(D \subseteq D^*\), and that the tails \(w_ n(z)\) have analytic continuations \(W_ n(z)\) to \(D^*\). In the paper are proved general results subsuming earlier results on analytic continuation. The structure of the main theorem is as follows: Given a continued fraction \(K^ \infty_{n=0} a_ n(z)/b_ n(z)\), being in a Near-set of a continued fraction in a certain Ref-set for a given domain \(D^*\). Then a sufficient condition for the sequence of \(W_ n(z)\)-modified approximants to be in Uniconv\((D^*_ 0)\) for some \(D^*_ 0 \subseteq D^*\) is given in terms of the parameter functions in the definitions of Ref and Near. In certain cases this result in turn leads to sufficient conditions for having continuous extension to the boundary of \(D^*_ 0\). Let it finally be mentioned that in the question of convergence of the modified approximants to the ``right function'' the author successfully uses the concept of general convergence, introduced by herself in Trans. Am. Math. Soc. 294, 477-485 (1986; Zbl 0587.40002)].
scientific article; zbMATH DE number 515198
scientific article