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

• Autorzy: Hajime Kaneko , Takeshi Kurosawa , Yohei Tachiya , Taka-aki Tanaka
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11J91: Transcendence theory of other special functions
• 11J81: Transcendence (general theory)
• 11J85: Algebraic independence; Gel\cprime fond's method

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 168
• Numer: 2
• Strony: 161-186

Daty

wydano

2015

Twórcy

autor

Hajime Kaneko

• Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki 350-0006, Japan

autor

Takeshi Kurosawa

• Department of Mathematical Information Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

autor

Yohei Tachiya

• Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

autor

Taka-aki Tanaka

• Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Identyfikatory

DOI

10.4064/aa168-2-5