Algebraic Independence of Mahler Functions via Radial Asymptotics

被引:13
|
作者
Brent, Richard P. [1 ]
Coons, Michael [2 ]
Zudilin, Wadim [2 ]
机构
[1] Australian Natl Univ, Inst Math Sci, GPO Box 4, Canberra, ACT 0200, Australia
[2] Univ Newcastle, Sch Math & Phys Sci, Callaghan, NSW 2308, Australia
基金
澳大利亚研究理事会;
关键词
GENERATING-FUNCTIONS; TRANSCENDENCE;
D O I
10.1093/imrn/rnv139
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behavior of a Mahler function f(z) as z goes radially to a root of unity to deduce algebraic independence results about the values of f(z) at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to F(z), the power series solution to the functional equation F(z) - (1 + z + z(2)) F (z(4)) + z(4)F (z(16))=0. Specifically, we prove that the functions F(z), F(z(4)), F' (z), and F' (z(4)) 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 a in the unit disc.
引用
收藏
页码:571 / 603
页数:33
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