Algebraic independence of the values of the Hecke–Mahler series and its derivatives at algebraic numbers

2018 ◽  
Vol 14 (09) ◽  
pp. 2369-2384 ◽  
Author(s):  
Taka-aki Tanaka ◽  
Yusuke Tanuma
Keyword(s):  
Unit Circle ◽  
Generating Function ◽  
Irrational Number ◽  
Algebraic Independence ◽  
Suitable Condition ◽  
Algebraic Numbers ◽  
Quadratic Irrational ◽  
Derivatives Of

We show that the Hecke–Mahler series, the generating function of the sequence [Formula: see text] for [Formula: see text] real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if [Formula: see text] is a quadratic irrational number satisfying a suitable condition.

DEGREE-ONE MAHLER FUNCTIONS: ASYMPTOTICS, APPLICATIONS AND SPECULATIONS

2020 ◽  
Vol 102 (3) ◽  
pp. 399-409
Author(s):  
MICHAEL COONS
Keyword(s):  
Unit Circle ◽  
Generating Function ◽  
Algebraic Independence ◽  
Roots Of Unity ◽  
Asymptotic Bounds ◽  
Mahler Function ◽  
Mahler Functions ◽  
Generic Points ◽  
Complete Characterisation

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

scholarly journals Algebraic independence properties of the Fredholm series

1978 ◽  
Vol 26 (1) ◽  
pp. 31-45 ◽  
Author(s):  
J. H. Loxton ◽  
A. J. van der Poorten
Keyword(s):  
Algebraic Independence ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Independence Properties ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF SERIES RELATED TO STERN'S SEQUENCE

2012 ◽  
Vol 08 (02) ◽  
pp. 361-376 ◽  
Author(s):  
PETER BUNDSCHUH
Keyword(s):  
Unit Disk ◽  
Generating Function ◽  
Algebraic Independence ◽  
Function Of Two Variables ◽  
Stern Sequence

In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211–217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental.

scholarly journals Short proofs of theorems of Lekkerkerker and Ballieu

1982 ◽  
Vol 5 (3) ◽  
pp. 609-612 ◽  
Author(s):  
Max Riederle
Keyword(s):  
Irrational Number ◽  
Quadratic Irrational ◽  
Accumulation Points

For any irrational numberξletA(ξ)denote the set of all accumulation points of{z:z=q(qξ−p),   p∈ℤ,   q∈ℤ−{0},   p   and   q   relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The setA(ξ)is discrete and does not contain zero if and only ifξis a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.

scholarly journals Hypertranscendance des systèmes aux différences diagonaux

2008 ◽  
Vol 144 (3) ◽  
pp. 565-581 ◽  
Author(s):  
Charlotte Hardouin
Keyword(s):  
Elliptic Curve ◽  
Linear Algebra ◽  
Galois Theory ◽  
Algebraic Independence ◽  
Difference Operators ◽  
Difference Systems ◽  
Derivatives Of

AbstractThis paper deals with criteria of algebraic independence for the derivatives of solutions of diagonal difference systems. The key idea consists in deriving from the commutativity of the differentiation and difference operators a sequence of iterated extensions of the original difference module, thereby setting the problem in the framework of difference Galois theory and finally reducing it to an exercise in linear algebra on the group of divisors of the involved elliptic curve or torus.

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