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${session.getAttribute("locale")}5Degree-one Mahler functions: asymptotics, applications and speculations
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k^{n}, where k is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over C(z). Finally, we discuss asymptotic bounds towards generic points on the unit circle.]]>Tue 10 May 2022 14:26:34 AEST]]>Transcendence of generating functions whose coefficients are multiplicative
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f : ℕ → K be a multiplicative function taking values in a field K of characteristic 0, and write F(z) = Σ_{n≥1}f(n)z^{n} ∈ K[[z]] for its generating series. If F(z) is algebraic, then either there is a natural number k and a periodic multiplicative function χ(n) such that f(n) = n^{k}χ(n) for all n or f(n) is eventually zero. In particular, the generating series of a multiplicative function taking values in a field of characteristic zero is either transcendental or rational. For K = ℂ, we also prove that if the generating series of a multiplicative function is D-finite, then it must either be transcendental or rational.]]>Sat 24 Mar 2018 08:44:34 AEDT]]>Regular sequences and the joint spectral radius
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k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the base-k logarithm of the joint spectral radius of any set of a special class of matrices determined by the k-kernel.]]>Sat 24 Mar 2018 07:30:39 AEDT]]>Growth degree classification for finitely generated semigroups of integer matrices
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A be a finite set of d x d matrices with integer entries and let m_{n}(Α) be the maximum norm of a product of n elements of A. In this paper, we classify gaps in the growth m_{n}(Α); specifically, we prove that lim_{n→∞}log m_{n}(A)/log n ∈ℤ≥₀⋃{∞}. This has applications to the growth of regular sequences as defined by Allouche and Shallit.]]>Sat 24 Mar 2018 07:10:14 AEDT]]>