Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime Kaneko, et al. "Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers." Acta Arithmetica 168.2 (2015): 161-186. <http://eudml.org/doc/279621>.

@article{HajimeKaneko2015,
abstract = {Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $∏_\{k=1 \atop U_\{d^k\}≠-a_i\}^\{∞\} (1 + (a_i)/(U_\{d^k\}))$ (i=1,...,m) or $∏_\{k=1 \atop V_\{d^k\}≠-a_i\}^\{∞\} (1 + (a_i)(V_\{d^k\})$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent.},
author = {Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka},
journal = {Acta Arithmetica},
keywords = {algebraic dependence; infinite products; Fibonacci numbers; Lucas numbers; Mahler's method},
language = {eng},
number = {2},
pages = {161-186},
title = {Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers},
url = {http://eudml.org/doc/279621},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Hajime Kaneko
AU - Takeshi Kurosawa
AU - Yohei Tachiya
AU - Taka-aki Tanaka
TI - Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 2
SP - 161
EP - 186
AB - Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $∏_{k=1 \atop U_{d^k}≠-a_i}^{∞} (1 + (a_i)/(U_{d^k}))$ (i=1,...,m) or $∏_{k=1 \atop V_{d^k}≠-a_i}^{∞} (1 + (a_i)(V_{d^k})$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent.
LA - eng
KW - algebraic dependence; infinite products; Fibonacci numbers; Lucas numbers; Mahler's method
UR - http://eudml.org/doc/279621
ER -