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Degree-one Mahler functions: asymptotics, applications and speculations. (English) Zbl 1461.11109

A (\(k\)-)Mahler function is a power series \(F(z)\in \mathbb C[[z]]\) solution of a non-trivial functional equation of the form \[ \sum_{j=0}^d a_j(z) F(z^{k^j})=0, \] where \(k\in \mathbb N\), \(k\ge 2\), \(d\ge 0\) and \(a_j(z)\in \mathbb C(z)\). Such a function is known is known to be in \(\mathbb C(z)\) or to have the circle \(\vert z\vert=1\) has a natural boundary. The integer \(d\) is called the degree of \(F(z)\) in this paper, and it is also called the order in the literature.
Mahler functions have been the object of intensive researches since Mahler first studied their Diophantine properties in 1929, in particular because of important connections with generating functions of automatic sequences. Simple examples are \(\sum_{n=0}^\infty z^{2^n}\) (for which \(d=2\) and \(k=2\)), and \(T_2(z):=\sum_{n=0}^\infty t(n)z^n\) where \(t(n)\) is the Thue-Morse (automatic) sequence defined on the alphabet \(\{1,-1\}\) by \(t_0=1\), \(t_{2n}=t_{2n+1}\) and \(t_{2n+1}=-t_n\). \(T_2(z)\) is a 2-Mahler function of order \(d=1\) because \(T_2(z)=\prod_{j=0}^\infty (1-z^{2^j})\). A straighforward generalization is \(T_k(z):=\prod_{j=0}^\infty (1-z^{k^j})\), obviously a \(k\)-Mahler function of order 1. More generally, \(k\)-Mahler functions of order 1 are of the form \(\prod_{j=0}^\infty r(z^{k^j})\) for some \(r(z)\in \mathbb C(z)\).
In this paper, the author first determines the asymptotic radial behavior \(k\)-Mahler functions of order 1 as \(z\to \xi\) along a ray starting from 0 where \(\xi\) is any primitive \(k^n\)-th root of unity. The result is too technical to be completely written in this review; it is a combination of a result of [N. G. Bruijn, Proc. Akad. Wet. Amsterdam 51, 659–669 (1948; Zbl 0030.34502)] on the radial behavior as \(z\to 1^-\) of the function \(T_k(z)\) and of a recent result of the author and J. P. Bell [Proc. Am. Math. Soc. 145, No. 3, 1061–1070 (2017; Zbl 1365.11092)] on the radial behavior as \(z\to 1^-\) of \(\prod_{j=0}^\infty p(z^{k^j})\) where \(p(z)\in \mathbb C[z]\) is such that \(p(0)=1\), \(p(1)\neq 0\). For \(T_k(z)\), de Bruijn’s results reads \[ T_k(z)=D_k(z)\cdot(1-z)^{1/2}k^{-\log_k(1-z)^2/2}(1+o(1)), \quad z\to 1^-, \] where \(D_k(z)\) is bounded away from 0 and infinity, real analytic and satisfies \(D_k(z)=D_k(z^k)\), and the main result (Theorem 3.4) is very similar.
As an application, the author then proves that \(T_k(z)\) and \(T_\ell(z)\) are algebraically independent over \(\mathbb C(z)\) when the integers \(k,\ell\ge 2\) are multiplicatively independent. He concludes with some generalizations of this result, where the notion of Mahler eigenvalues (introduced in the above mentioned paper with Bell) is used.

MSC:

11J91 Transcendence theory of other special functions
11B85 Automata sequences
30B10 Power series (including lacunary series) in one complex variable
68R15 Combinatorics on words
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References:

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