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

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.