## Functiones et Approximatio Commentarii Mathematici

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

#### 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

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

#### 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

#### References

• P.-G. Becker, T. Töpfer, Transcendency results for sums of reciprocals of linear recurrences, Math. Nachr. 168 (1994) 5–17.
• P. Bundschuh, K. Väänänen, Some arithmetical results on reciprocal sums of certain Fibonacci-type numbers, Southeast Asian Bull. Math. (to appear).
• P. Bundschuh, K. Väänänen, Algebraic independence of certain Mahler functions and of their values, J. Aust. Math. Soc. 98 (2015) 289–310.
• P. Bundschuh, K. Väänänen, Algebraic independence of reciprocal sums of certain Fibonacci-type numbers, arXiv:1403.5510v1 [math. NT] (2014).
• D. Duverney, T. Kanoko, T. Tanaka, Transcendence of certain reciprocal sums of linear recurrences, Monatsh. Math. 137 (2002) 115–128.
• T. Kanoko, T. Kurosawa, I. Shiokawa, Transcendence of reciprocal sums of binary recurrences, Monatsh. Math. 157 (2009) 323–334.
• K. Nishioka, Mahler Functions and Transcendence, LNM 1631 (Springer, Berlin et al., 1996).
• K. Nishioka, Algebraic independence of reciprocal sums of binary recurrences, Monatsh. Math. 123 (1997) 135–148.
• K. Nishioka, Algebraic independence of reciprocal sums of binary recurrences II, Monatsh. Math. 136 (2002) 123–141.
• K. Nishioka, T. Tanaka, T. Toshimitsu, Algebraic independence of sums of reciprocals of the Fibonacci numbers, Math. Nachr. 202 (1999) 97–108.