Levels of multi-continued fraction expansion of multi-formal Laurent series

AbstractThe multi-continued fraction expansion C(r̲) of a multi-formal Laurent series r̲ is a sequence pair (h̲,a̲) consisting of an index sequence h̲ and a multi-polynomial sequence a̲. We denote the set of the different indices appearing infinitely many times in h̲ by H∞, the set of the different indices appearing in h̲ by H+, and call |H∞| and |H+| the first and second levels of C(r̲), respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of r̲ are determined by H∞ (H+), and how the components are linearly dependent on the mentioned basis