Infinite series representations for complex numbers (Q1115898)

With the help of certain lattices in \({\mathbb{Q}}(\sqrt{-m})\) and suitable chosen fundamental sets generalizations of both Engel and Sylvester series can be obtained for complex numbers. This leads to representations of complex numbers as a series of reciprocals of algebraic integers. A typical result reads as follows: Let \(m=1\) or 2, then every complex number z has a representation in the form \(z=a_ 0+\sum^{\infty}_{j=1}a_ j^{-1},\quad a_ j\in {\mathbb{Z}}[\sqrt{- m}],\) where \(| a_{n+1}| \geq \gamma^{-1}| a_ n|^ 2-| a_ n| -\gamma,\quad 2\gamma =\sqrt{m+1}.\)