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DEGREE-ONE MAHLER FUNCTIONS: ASYMPTOTICS, APPLICATIONS AND SPECULATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 3 / December 2020
- Published online by Cambridge University Press:
- 05 February 2020, pp. 399-409
- Print publication:
- December 2020
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- Article
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We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as
$z$ approaches roots of unity of degree 
$k^{n}$, 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 
$\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.
TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 95 / Issue 3 / June 2017
- Published online by Cambridge University Press:
- 13 March 2017, pp. 393-399
- Print publication:
- June 2017
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- Article
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In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if
$F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then 
$F(z)$ is either rational or it is transcendental over the field of meromorphic functions.
