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ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 98 / Issue 3 / June 2015
- Published online by Cambridge University Press:
- 11 November 2014, pp. 289-310
- Print publication:
- June 2015
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- Article
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This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations
$af(z^{r})=f(z)+R(z)$, where 
$a$ is a nonzero complex number, 
$r$ an integer greater than 1, and 
$R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.
