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

Abstract

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].