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On the domain of convergence and poles of complex J-fractions

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1998-05
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Elsevier
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Abstract
Consider the infinite $J$-fraction $$\cfrac a_0 \\ z-b_0-\cfrac a_1 \\ z-b_1-\cfrac a_2 \\ z-b_2-{\lower6pt\hbox{$\ddots$}}\endcfrac$$ where $a_n\in{\bf C}\sbs\{0\},\ b_n\in{\bf C}$. Under very general conditions on the coefficients $\{a_n\},\ \{b_n\}$, we prove that this continued fraction coverges to a meromorphic function in ${\bf C}\sbs{\bf R}$. Such conditions hold, in particular, if $\lim_n{\rm Im}(a_n)=\lim_n{\rm Im}(b_n)=0$ and $\sum_{n\ge0}(1/ _n =\infty$ (or $\sum_{n\ge0}( _n _na_{n+1} =\infty)$. The poles are located in the point spectrum of the associated tridiagonal infinite matrix and their order determined in terms of the asymptotic behavior of the zeros of the denominators of the corresponding partial fractions.
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24 pages, no figures.-- MSC2000 code: 30B60.
MR#: MR1616769 (99b:30003)
Zbl#: Zbl 0909.30002
Keywords
Continued fractions, J-fractions, Tridiagonal infinite matrices, Asymptotic behaviour of poles
Bibliographic citation
Journal of Approximation Theory, 1998, vol. 93, n. 2, p. 177-200