scholarly journals Becker’s conjecture on Mahler functions

2019 ◽  
Vol 372 (5) ◽  
pp. 3405-3423
Author(s):  
Jason P. Bell ◽  
Frédéric Chyzak ◽  
Michael Coons ◽  
Philippe Dumas
Keyword(s):  
Mahler Functions

scholarly journals On linear independence measures of the values of Mahler functions

2018 ◽  
Vol 148 (6) ◽  
pp. 1297-1311 ◽  
Author(s):  
Keijo Väänänen ◽  
Wen Wu
Keyword(s):  
Functional Equations ◽  
Linear Independence ◽  
Infinite Sequence ◽  
Padé Approximation ◽  
Pade Approximation ◽  
Mahler Functions

We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.

Algebraic theory of difference equations and Mahler functions

2012 ◽  
Vol 84 (3) ◽  
pp. 245-259
Author(s):  
Kumiko Nishioka ◽  
Seiji Nishioka
Keyword(s):  
Difference Equations ◽  
Algebraic Theory ◽  
Mahler Functions

scholarly journals Algebraic independence of certain Mahler numbers

2016 ◽  
Vol 12 (08) ◽  
pp. 2159-2166
Author(s):  
Keijo Väänänen
Keyword(s):  
Unit Disk ◽  
Generating Functions ◽  
Special Class ◽  
Algebraic Independence ◽  
Open Unit ◽  
Open Unit Disk ◽  
Algebraic Point ◽  
Mahler’S Method ◽  
Mahler Functions ◽  
Independence Results

In this note, we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue–Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.

scholarly journals Algebraic Independence of Mahler Functions via Radial Asymptotics

2015 ◽  
pp. rnv139 ◽  
Author(s):  
Richard P. Brent ◽  
Michael Coons ◽  
Wadim Zudilin
Keyword(s):  
Algebraic Independence ◽  
Mahler Functions

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

2014 ◽  
Vol 98 (3) ◽  
pp. 289-310 ◽  
Author(s):  
PETER BUNDSCHUH ◽  
KEIJO VÄÄNÄNEN
Keyword(s):  
Rational Function ◽  
Complex Number ◽  
Functional Equations ◽  
Algebraic Independence ◽  
Lucas Numbers ◽  
Mahler Functions ◽  
Independence Results ◽  
Nonzero Complex Number ◽  
Fibonacci And Lucas Numbers ◽  
Reciprocal Sums

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.

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