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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
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