Sequences of level 1,2,3, ... ,k, ...

Sequences of numbers (either natural integers, or integers or rational) of level k epsilon IN have been defined in [FS06] as the sequences which can be computed by deterministic pushdown automata of level k. We extend this definition to sequences of words indexed by words. We give characterisations of these sequences in terms of "higher-order" L-systems. In particular sequences of rational numbers of level 3 are characterised by polynomial recurrences (which generalize the P-recurrent sequences studied in [Sta80]). The equality problem for sequences of rational numbers of level 3 is shown decidable.