Functiones et Approximatio Commentarii Mathematici

Algebraic independence of reciprocal sums of powers of certain Fibonacci-type numbers

Peter Bundschuh and Keijo Väänänen

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Abstract

The Fibonacci-type numbers in the title look like $R_n=g_1\gamma_1^n+ g_2\gamma_2^n$ and $S_n=h_1\gamma_1^n+h_2\gamma_2^n$ for any $n\in\mathbb{Z}$, where the $g$'s, $h$'s, and $\gamma$'s are given algebraic numbers satisfying certain natural conditions. For fixed $k\in\mathbb{Z}_{>0}$, and for fixed non-zero periodic sequences $(a_h),(b_h),(c_h)$ of algebraic numbers, the algebraic independence of the series \[ \sum_{h=0}^\infty \frac{a_h}{\gamma_1^{kr^h}}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{b_h}{(R_{kr^h+\ell})^m}\,, \quad {\sum_{h=0}^\infty}\,\strut' \frac{c_h}{(S_{kr^h+\ell})^m} \qquad \big((\ell,m,r)\in\mathbb{Z}\times \mathbb{Z}_{>0}\times\mathbb{Z}_{>1}\big) \] is studied. Here the main tool is Mahler's method which reduces the investigation of the algebraic independence of numbers (over $\mathbb{Q}$) to that of functions (over the rational function field) if they satisfy certain types of functional equations.

Article information

Source
Funct. Approx. Comment. Math., Volume 53, Number 1 (2015), 47-68.

Dates
First available in Project Euclid: 28 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.facm/1443444850

Digital Object Identifier
doi:10.7169/facm/2015.53.1.4

Mathematical Reviews number (MathSciNet)
MR3402772

Zentralblatt MATH identifier
06862317

Subjects
Primary: 11J91: Transcendence theory of other special functions
Secondary: 11J81: Transcendence (general theory) 39B32: Equations for complex functions [See also 30D05]

Keywords
algebraic independence of numbers Mahler's method algebraic independence of functions

Citation

Bundschuh, Peter; Väänänen, Keijo. Algebraic independence of reciprocal sums of powers of certain Fibonacci-type numbers. Funct. Approx. Comment. Math. 53 (2015), no. 1, 47--68. doi:10.7169/facm/2015.53.1.4. https://projecteuclid.org/euclid.facm/1443444850


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References

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