https://nova.newcastle.edu.au/vital/access/ /manager/Index ${session.getAttribute("locale")} 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:26658 Wed 11 Apr 2018 17:08:06 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:23973 2)F(z⁴) +z⁴F(z¹⁶)=0. Specifically, we prove that the functions F(z), F(z⁴), F′(z), and F′(z⁴) are algebraically independent over ℂ(z). An application of a celebrated result of Ku. Nishioka then allows one to replace ℂ(z) by ℚ when evaluating these functions at a nonzero algebraic number α in the unit disc.]]> Wed 11 Apr 2018 15:22:41 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:30913 z, but also over Cz(M), where (M) is the set of meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.]]> Wed 11 Apr 2018 15:20:26 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:39110 kⁿ, 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 ]]>