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

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H∞, the set of the different indices appearing in by H+, and call H∞ and H+ the first and second levels of , 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 are determined by H∞ (H+), and how the components are linearly dependent on the mentioned basis.