On convergence criteria for branched continued fraction

Abstract

The starting point of the present paper is a result by E.A. Boltarovych (1989) on convergence regions, dealing with branched continued fraction $$\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{1}{\atop+}\ldots{\atop+}\sum_{i_n=1}^N\frac{a_{i(n)}}{1}{\atop+}\ldots,$$ where $|a_{i(2n-1)}|\le\alpha/N,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and $|a_{i(2n)}|\le r/(N-1),$ $i_{2n}\ne j_{2n},$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ where $N>1$ and $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions. In the present paper conditions for these regions are replaced by $\sum_{i_1=1}^N|a_{i(1)}|\le\alpha(1-\varepsilon),$ $\sum_{i_{2n+1}=1}^N|a_{i(2n+1)}|\le\alpha(1-\varepsilon),$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R$ and $\sum_{i_{2n}\in\{1,2,\ldots,N\}\backslash\{j_{2n}\}}|a_{i(2n)}|\le r,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ where $\varepsilon,$ $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions, and better convergence speed estimates are obtained.