Algebraic theory of difference equations and Mahler functions

On the algebraic relations between Mahler functions

“…We also compute the Galois group of the "direct sum" of the Baum-Sweet and of the Rudin-Shapiro equations (via the Goursat-Kolchin-Ribet lemma), which turns out to be equal to the direct product of the Galois groups of the Baum-Sweet and of the Rudin-Shapiro equations. For instance, this gives a Galoisian proof of the following result obtained by Nishioka and Nishioka in [NN12]: if we let f 1 (z) = f (z) (resp. g(z)) be the generating series of the Rudin-Shapiro (resp.…”

“…A number of authors have studied the algebraic relations between the generating series of certain p-automatic sequences. For instance, the generating series of the so-called Baum-Sweet and Rudin-Shapiro sequences (see sections 9.1 and 9.2) were studied by Nishioka and Nishioka in [NN12]: they are algebraically independent over Q(z). 1 It turns out that the generating series f (z) = k≥0 s k z k of any pautomatic sequence (s k ) k≥0 ∈ Q N (and, actually, of any p-regular sequence) satisfies…”

Note on the Stern-Brocot sequence, some relatives, and their generating power series

“…In a recent paper, K. Nishioka and S. Nishioka [9] proved much stronger results on Φ and Ψ. Namely, they obtained the algebraic independence of all four functions Φ(z), Φ(−z), Ψ(z), Ψ(z 2 ) not only over C(z) but over any difference field extension of valuation ring type over C(z) under the transformation z → z 2 (compare [9,Theorem 9]). This notion, too combersome to be fully quoted here, has been introduced by S. Nishioka in his deep work [10, Definition 1] on solvability of certain classes of difference equations.…”

“…This notion, too combersome to be fully quoted here, has been introduced by S. Nishioka in his deep work [10, Definition 1] on solvability of certain classes of difference equations. His Theorem 2 and Proposition 5 are the key tools of the proofs in [9].…”