## A problem about Mahler functions.(English)Zbl 1432.11086

Summary: Let $$K$$ be a field of characteristic zero and $$k$$ and $$l$$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by J. H. Loxton and A. J. van der Poorten [J. Reine Angew. Math. 330, 159–172 (1982; Zbl 0468.10019); ibid. 392, 57–69 (1988; Zbl 0656.10033)] during the Eighties: a power series $$F(z)\in K[[z]]$$ satisfies both a $$k$$- and a $$l$$-Mahler-type functional equation if and only if it is a rational function.

### MSC:

 11J81 Transcendence (general theory) 11B85 Automata sequences 65Q20 Numerical methods for functional equations

Mahler functions

### Citations:

Zbl 0468.10019; Zbl 0656.10033
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