Convergence properties of a class of \(\hat J\)-fractions (Q1114877)

In this paper the convergence behaviour of so-called \(\hat J\)-fractions is studied. The partial denominators resp. numerators are given by \(z+b_ n\) \((b_ n\) complex) and \(-\{a_ n(z)\}^ 2\) \((a_ 1(z)=1\), \(a_ n(z)=c_ nz^{\delta_ n}\) with \(c_ n\) complex and \(\delta_ n\in \{0,1\})\) resp. If all \(\delta_ n\) are zero, the ordinary J- fraction is recovered. First conditions are given under which the \(\hat J\)-fraction is positive definite (for real \(\hat J-\)fractions these conditions are necessary and sufficient). Then the convergence behaviour of positive definite real \(\hat J\)-fraction with the special choice \(a_{2n}(z)=c_{2n}\), \(a_{2n+1}(z)=c_{2n+1}z\), \(c_ n\) real, is treated in detail. Using the approach through linear fractional transformations, conditions are given under which we have the limit-point case or the limit-circle case. Finally conditions for uniform convergence over bounded closed regions in the upper or lower half plane (for real \(\hat J\)-fractions only) or outside a disc with finite positive radius are given. The paper gives an interesting contribution to the analytic theory of continued fractions.