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Transcendence tests for Mahler functions. (English) Zbl 1365.11092

The authors propose two tests for transcendence of Mahler functions \(F(z)\). The first one is a “quick” test based on the notion of the eigenvalue \(\lambda_{F}\) of a Mahler function. The second test is a little bit slower, however more general and depends on the rank of a certain Hankel matrix determined by the initial coefficients of \(F(z)\). The eigenvalue \(\lambda_{F}\) has properties which are similar to Perron-Frobenius theory for real matrices with non-negative entries. As a nice example, a short proof for the transcendence of the generating function of the Thue-Morse sequence is presented.

MSC:

11J91 Transcendence theory of other special functions
39A06 Linear difference equations
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