https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:14415 Wed 11 Apr 2018 15:47:25 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:34337 F of a Mahler function F(z) and develop a quick test for the transcendence of F(z) over ℂ(z), which is determined by the value of the eigenvalue λ_F. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of F(z). We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.]]> Wed 04 Sep 2019 10:06:10 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:31335 f : ℕ → K be a multiplicative function taking values in a field K of characteristic 0, and write F(z) = Σ_n≥1f(n)zⁿ ∈ 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^kχ(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 ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:20778 0 such that ∣f(n)∣ > c log n infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.]]> Sat 24 Mar 2018 08:06:02 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:28287 Sat 24 Mar 2018 07:41:23 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23925 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 ]]>