Two-dimensional multiplicative relations involving coefficients and roots of generalized Fibonacci polynomials

Abstract Let $$\alpha $$ α be the positive real root of the k th generalized Fibonacci polynomial $$\begin{aligned} f_k(X) = X^k - X^{k-1} - \cdots - X - 1. \end{aligned}$$ f k ( X ) = X k - X k - 1 - ⋯ - X - 1 . In this work, we show that there are only finitely many primitive nontrivial multiplicative relations of the form $$g_{k,a,b}(\alpha )^r \alpha ^s = A$$ g k , a , b ( α ) r α s = A , where $$g_{k,a,b}(\alpha ) = \frac{a\alpha + b}{f_k'(\alpha )}$$ g k , a , b ( α ) = a α + b f k ′ ( α ) , with $$k \geqslant 3$$ k ⩾ 3 , $$a,b,r,s \in \mathbb {Z}$$ a , b , r , s ∈ Z , and $$A \in \mathbb {Q}$$ A ∈ Q .