No label defined (Q6645081)

Let \(\{F_n\}\) be the Fibonacci sequence; \(F_0=0,F_1=1\) and the recurrence \(F_n=F_{n-1}+F_{n-2}\). These are given by Binet's formula \(F_n=\frac{\alpha^n-\beta^n}{\sqrt{5}}\), where \(\alpha=\frac{1+\sqrt{5}}{2}\) and \(\beta=\frac{1-\sqrt{5}}{2}\) are the roots of the characteristic polynomial \(x^2-x-1\).\N\NFor rational polynomials \(P_0(n), P_1(n)\), the authors consider the integrality of the sequence \(w_n = P_0(n)F_n + P_1(n)F_{n-1}\). Since\N\[\Nw_n =\frac{1}{\sqrt{5}}\left(P_0(n)+\frac{P_1(n)}{\alpha} \right)\alpha^n+\frac{1}{\sqrt{5}}\left(P_0(n)+\frac{P_1(n)}{\beta} \right)\beta^n\N\]\NThe last expression shows that the corresponding characteristic polynomial of the sequence \(\{w_n\}\) has zeros \(\alpha\) and \(\beta\) with certain multiplicities.\N\NThus the authors consider the sequence\N\[\N w_n = q_\alpha(n)\alpha^n + q_\beta(n)\beta^n\tag{2.2}\N\]\Nwhere \(q_\alpha(x), q_\beta(x)\in \mathbb{C}[x]\) with degrees \(d_\alpha, d_\beta\). Then from the characteristic polynomial \(c_w(x)=(x-\alpha)^{d_\alpha}(x-\beta)^{d_\beta}\), they factor out \((x^2-x-1)^{\min\{d_\alpha,d_\beta\}}\). Then two cases of equal and unequal degrees are considered. E.g. Theorem 2 determines the case where \(w_n = (an+b)F_n + (cn+d)F_{n-1}\) is an integer sequence: if \(d\) is an integer, then values are explicitly given for integrality.