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1
Content available remote

Linear independence of certain Lambert and allied series

100%
Bundschuh P., Väänänen K.
Acta Arithmetica
|
2005
|
tom 120
|
nr 2
197-209
2
Content available remote

Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II

100%
Bundschuh P., Väänänen K.
Acta Arithmetica
|
2015
|
tom 167
|
nr 3
239-249
EN
This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic α with 0 < |α| < 1, the field ℚ(F(α),F(α⁴),G(α),G(α⁴)) has transcendence degree 3 over ℚ.
3
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Transcendence results on the generating functions of the characteristic functions of certain self-generating sets

100%
Bundschuh P., Väänänen K.
Acta Arithmetica
|
2014
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tom 162
|
nr 3
273-288
EN
This article continues two papers which recently appeared in this same journal. First, Dilcher and Stolarsky [140 (2009)] introduced two new power series, F(z) and G(z), related to the so-called Stern polynomials and having coefficients 0 and 1 only. Shortly later, Adamczewski [142 (2010)] proved, inter alia, that G(α),G(α⁴) are algebraically independent for any algebraic α with 0 < |α| < 1. Our first key result is that F and G have large blocks of consecutive zero coefficients. Then, a Roth-type argument shows that F(a/b) and G(a/b), for any (a,b) ∈ ℤ × ℕ with 0 < |a| < √b, are transcendental but not U-numbers. Moreover, reasonably good upper bounds for the irrationality exponent of these numbers are obtained. Another main result for which an elementary (or poor men's) proof is presented concerns the algebraic independence of F(z),F(z⁴) over ℂ(z) leading to the F-analogue of Adamczewski's above-mentioned theorem.
4
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Arithmetical properties of the values of functions satisfying certain functional equations of Poincaré

81%
Amou M., Katsurada M., Väänänen K.
Acta Arithmetica
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2001
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tom 99
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nr 4
389-407
5
Content available remote

On Siegel-Shidlovskii's theory for q-difference equations

81%
Amou M., Matala-aho T., Väänänen K.
Acta Arithmetica
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2007
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tom 127
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nr 4
309-335
6
Content available remote

On the non-quadraticity of values of the q-exponential function and related q-series

81%
Krattenthaler C., Rochev I., Väänänen K., Zudilin W.
Acta Arithmetica
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2009
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tom 136
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nr 3
243-269
7
Content available remote

On linear forms of a certain class of G-functions and p-adic G-functions

60%
Väänänen K.
Acta Arithmetica
|
1980
|
tom 36
|
nr 3
273-295
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