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Some results on separate convergence of continued fractions. (English) Zbl 0711.30006

Computational methods and function theory, Proc. Conf., Valparaíso/Chile 1989, Lect. Notes Math. 1435, 191-200 (1990).
[For the entire collection see Zbl 0702.00014.]
Let (1) \(K(a_ n(z)/b_ n(z))\) be a limit periodic continued fraction with lim \(a_ n(z)=a(z)\), lim \(b_ n(z)=b(z)\). Let \(A_ n(z)\) and \(B_ n(z)\) be the numerators and denominators, respectively, of the nth approximant of (1), as normalized by the standard recursion relations. Let \(x_ 1(z)\) and \(x_ 2(z)\) be solutions of \(w^ 2+b(z)w-a(z)=0\). The main theorem of the article can then be stated as follows: Let \(\Delta\) be a region in the complex plane such that for \(z\in \Delta\) the three conditions
(a) the functions \(a_ n(z)\), \(b_ n(z)\), a(z), b(z) are holomorphic,
(b) \(| x_ 1(z)/x_ 2(z)| <1,\)
(c) \(\sum | a_ n-a|\), \(\sum | b_ n-b|\) both converge,
are satisfied. Then \[ \lim \frac{A_ n(z)}{(-x_ 2(z))^{n+1}}\text{ and } \lim \frac{B_ n(z)}{(-x_ 2(z))^{n+1}} \] both exist and are holomorphic for all \(z\in \Delta\). Applications are made to regular C- functions (these results were initially proved by Śleszyński in 1888) and general T-fractions (overlapping earlier results of Waadeland and Waadeland and Thron).
Reviewer: W.J.Thron

MSC:

30B70 Continued fractions; complex-analytic aspects
40A15 Convergence and divergence of continued fractions

Citations:

Zbl 0702.00014