scholarly journals Algebraic independence results for values of theta-constants

2015 ◽  
Vol 52 (1) ◽  
pp. 7-27 ◽  
Author(s):  
Carsten Elsner
Keyword(s):  
Algebraic Independence ◽  
Independence Results

Algebraic Independence Results Related to 〈q, r〉-Number Systems

2006 ◽  
Vol 147 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Shin-ichiro Okada ◽  
Iekata Shiokawa
Keyword(s):  
Algebraic Independence ◽  
Number Systems ◽  
Independence Results

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.

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.

ARITHMETIC PROPERTIES OF INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

2016 ◽  
Vol 93 (3) ◽  
pp. 375-387 ◽  
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.

Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$

2018 ◽  
Vol 123 (2) ◽  
pp. 249-272
Author(s):  
Carsten Elsner ◽  
Yohei Tachiya
Keyword(s):  
Differential Operator ◽  
Theta Function ◽  
Algebraic Number ◽  
Complex Number ◽  
Algebraic Independence ◽  
Complex Variables ◽  
Jacobi Theta Function ◽  
Upper Half Plane ◽  
Independence Results ◽  
Special Case

In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{\nu =-\infty }^{\infty } e^{\pi i\nu ^2\tau + 2\pi i\nu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{\nu =1}^{\infty } e^{\pi i\nu ^2 \tau } \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $\theta (z|\tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $\theta _3(\tau )$. For example, the three values $\theta _3(\tau )$, $\theta _3(n\tau )$, and $D\theta _3(\tau )$ are algebraically independent over $\mathbb{Q} $ for any τ such that $q=e^{\pi i\tau }$ is an algebraic number, where $n\geq 2$ is an integer and $D:=(\pi i)^{-1}{d}/{d\tau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $\theta _3(\tau )$ and $\theta _3(2^m\tau )$ for $m\geq 1$. As an application of our main theorem, the algebraic dependence over $\mathbb{Q} $ of the three values $\theta _3(\ell \tau )$, $\theta _3(m\tau )$, and $\theta _3(n\tau )$ for integers $\ell ,m,n\geq 1$ is also presented.

scholarly journals Algebraic independence results on the generating Lambert series of the powers of a fixed integer

2015 ◽  
Vol Volume 38 ◽  
Author(s):  
Peter Bundschuh ◽  
Keijo Väänänen
Keyword(s):  
Functional Equations ◽  
Algebraic Independence ◽  
Main Tool ◽  
Interesting Case ◽  
Lambert Series ◽  
Fixed Integer ◽  
International Audience ◽  
Independence Results ◽  
The One ◽  
Algebraic Independence Of Numbers

International audience In this paper, the algebraic independence of values of the functionG d (z) := h≥0 z d h /(1 − z d h), d > 1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more general functions. The main tool is Mahler's method reducing the investigation of the algebraic independence of numbers (over Q) to the one of functions (over the rational function field) if these satisfy certain types of functional equations.

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