Interpolation polynomials associated to linear recurrences (Q6542393)

Suppose that \(\{G_n\}\) is an arbitrary real linear recurrence of order \(k\). In this paper, the authors study the properties of the interpolation polynomials \(P_m(x)\) induced by the points \((t, G_t)\) with\N\[\Nt=n_0, n_1=n_0+1, n_2=n_0+2, \ldots, n_m=n_0+m\N\]\Ngiven in the planar Cartesian system for \(n_0 \in \mathbb{Z}\) and \(m \in \mathbb{N}\). To find \(P_m(x)\), they apply Newton's divided differences method. So they show that such polynomials can be obtained by using a new approach concentrating on the inner structure of the explicit formula of the recurrence.\N\NMore precisely, the main result of the paper is an explicit formula depends on the explicit formula of \(G_n\) and on the finite difference sequence of a specific sequence. It makes it possible to study the interpolation polynomials essentially by the zeros of the characteristic polynomial of \(\{G_n\}\). During the investigations, they also developed certain formulas related to the finite differences.