منابع مشابه
Elliptic Functions and Transcendence
Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and e which are related with the classical exponential function, it turns out that elliptic functions are required (so far – this should not last for ever!) to prove transcendence results and ge...
متن کاملBecker's conjecture on Mahler functions
In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be...
متن کاملAuxiliary functions in transcendence proofs
We discuss the role of auxiliary functions in the development of transcendental number theory. Earlier auxiliary functions were completely explicit (§ 1). The earliest transcendence proof is due to Liouville (§ 1.1), who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in o...
متن کاملA Problem about Mahler Functions
— Let K be a field of characteristic zero and k and l be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during the Eighties: a power series F (z) ∈ K[[z]] satisfies both a kand a l-Mahler type functional equation if and only if it is a rational function.
متن کامل7 Elliptic Functions and Transcendence
Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and eπ that are related to the classical exponential function, it turns out that elliptic functions are required (so far, this should not last forever!) to prove transcendence results and get a ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2016
ISSN: 0002-9939,1088-6826
DOI: 10.1090/proc/13297