An approach to Zaremba's conjecture throughout Fibonacci sequences (Q6619718)

Define the Fibonacci polynomials \(F_n(a)\) by the recurrence relation \(F_n(a)=a F_{n-1}(a)+F_{n-2}(a)\) (\(n\ge 2\)) with \(F_0(a)=0\) and \(F_1(a)=1\). When \(a=1\), \(F_n=F_n(1)\) are Fibonacci numbers. Recently in [Exp. Math., Latest articles; https://doi.org/10.1080/10586458.2023.2293285], the first author and Komatsu proved that for \(1\le a,b\le 5\) Zaremba conjecture is valid for 1) \(F_{n+1}(a)F_{m+1}(b)+F_n(a)F_m(b)\) (\(n\ge 1\), \(m\ge 1\)), 2) \(F_{n+1}(a)^2 F_{m+1}(b)+F_n(a)^2 F_{m-1}(b)+2 F_{n+1}(a)F_n(a)F_m(b)\) (\(n\ge 1\), \(m\ge 2\)) and their \(2^n\)-th powers except for \(a=5\) or \(b=5\). According to \textit{M. Drmota} [in: Applications of Fibonacci numbers. Volume 5: Proceedings of the fifth international conference on Fibonacci numbers and their applications, University of St. Andrews, Scotland, July 20-24, 1992. Dordrecht: Kluwer Academic Publishers. 185--197 (1993; Zbl 0805.11009)], Zaremba's conjecture holds for \(F_{n+1}+1\), \(F_{n+1}(2)+1\) and \(F_{n+1}(4)+1\) for certain \(n\)'s satisfying the congruences. In this paper, Zaremba's conjecture is examined in relation to the Fibonacci sequences as complementary results to those in the previous two papers. As complementary results to the first one, the results on sum, square, product and cube of Fibonacci sequences are given. As complementary results to the second one, the results for \((F_{n+1}(a)+1)^2+(F_n(a)+1)^2\), \(F_{2 n+1}(a)+2 F_{n+1}(a)+F_n(a)+1\), \(y(F_{n+1}(a)+1)^2+2(F_n(a)+1)(F_{n+1}(a)+1)\), \(y(F_{n+1}(a)+1)^2+(F_{n+1}(a)+1)(2 F_n(a)+1)\) and \((F_{n+1}(a)+1)(y F_{n+1}(a)+y+1)\) are shown.