Algebraic independence results for the values of certain Mahler functions and their application to infinite products

Takeshi Kurosawa; Yohei Tachiya; Taka-aki Tanaka

doi:10.1007/s00605-014-0617-3

Algebraic independence results for the values of certain Mahler functions and their application to infinite products

Monatshefte für Mathematik ◽

10.1007/s00605-014-0617-3 ◽

2014 ◽

Vol 174 (1) ◽

pp. 77-104

Author(s):

Takeshi Kurosawa ◽

Yohei Tachiya ◽

Taka-aki Tanaka

Keyword(s):

Algebraic Independence ◽

Infinite Products ◽

Mahler Functions ◽

Independence Results

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Algebraic independence of certain Mahler numbers

International Journal of Number Theory ◽

10.1142/s1793042116501293 ◽

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.

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ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788714000524 ◽

2014 ◽

Vol 98 (3) ◽

pp. 289-310 ◽

Cited By ~ 1

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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ARITHMETIC PROPERTIES OF INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715001550 ◽

2016 ◽

Vol 93 (3) ◽

pp. 375-387 ◽

Cited By ~ 2

Author(s):

PETER BUNDSCHUH ◽

KEIJO VÄÄNÄNEN

Keyword(s):

Algebraic Independence ◽

Cyclotomic Polynomials ◽

Infinite Products ◽

Arithmetic Properties ◽

Independence Results

We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.

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