Transcendence tests for Mahler functions
Jason P. Bell, Michael James Coons · Proceedings of the American Mathematical Society · 2016
We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue λ F \lambda _F of a Mahler function F ( z ) F(z) and develop a quick test for the transcendence of F ( z ) F(z) over C ( z ) \mathbb {C}(z) , which is determined by the value of the eigenvalue λ F \lambda _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 ) F(z) . We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given.