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

Let α α be the positive real root of the k th generalized Fibonacci polynomial aligned fₖ(X) = Xᵏ - Xᵏ⁻¹ - ⋯ - X - 1. 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 gk,a,b(α )ʳ α ˢ = A g k , a , b ( α ) r α s = A , where gk,a,b(α ) = aα + b/fₖ'(α ) g k , a , b ( α ) = a α + b f k ′ ( α ) , with k 3 k ⩾ 3 , a,b,r,s ∈ Z a , b , r , s ∈ Z , and A ∈ Q A ∈ Q .