The proof of a formula concerning the asymptotic behavior of the reciprocal sum of the square of multiple-angle Fibonacci numbers

Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F22%3A50018995" target="_blank" >RIV/62690094:18470/22:50018995 - isvavai.cz</a>
Výsledek na webu
<a href="https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-022-02755-7" target="_blank" >https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-022-02755-7</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1186/s13660-022-02755-7" target="_blank" >10.1186/s13660-022-02755-7</a>

Jazyk výsledku
angličtina
Název v původním jazyce
The proof of a formula concerning the asymptotic behavior of the reciprocal sum of the square of multiple-angle Fibonacci numbers
Popis výsledku v původním jazyce
Let (F-n)(n) be the Fibonacci sequence defined by Fn+2 = Fn+1 + F-n with F-0 = 0 and F-1 = 1. In this paper, we prove that for any integer m >= 1 there exists a positive constant C-m for which lim(n ->infinity){(Sigma(infinity)(k=n)1/F-mk(2))(-1) - (F-mn(2)-F-m(n-1)(2) + (-1)C-mn(m))} = 0. Furthermore, we show that C-m tends to 2/5 as m ->infinity (indeed, we provide quantitative versions of the previous results as well as an explicit form for C-m). This confirms some questions proposed by Lee and Park [J. Inequal. Appl. 2020(1):91 2020].
Název v anglickém jazyce
The proof of a formula concerning the asymptotic behavior of the reciprocal sum of the square of multiple-angle Fibonacci numbers
Popis výsledku anglicky
Let (F-n)(n) be the Fibonacci sequence defined by Fn+2 = Fn+1 + F-n with F-0 = 0 and F-1 = 1. In this paper, we prove that for any integer m >= 1 there exists a positive constant C-m for which lim(n ->infinity){(Sigma(infinity)(k=n)1/F-mk(2))(-1) - (F-mn(2)-F-m(n-1)(2) + (-1)C-mn(m))} = 0. Furthermore, we show that C-m tends to 2/5 as m ->infinity (indeed, we provide quantitative versions of the previous results as well as an explicit form for C-m). This confirms some questions proposed by Lee and Park [J. Inequal. Appl. 2020(1):91 2020].

Projekt
—
Návaznosti
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění
2022
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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