The Power Sums Involving Fibonacci Polynomials and Their Applications

Abstract

The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham.

On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of Σn k=0 xkW2

“…Name of sequence sums of second powers sums of third powers sums of powers Generalized Fibonacci [18,19,20,21,22,23] [ 24,25,26,27,28] [ 29,30,31] Generalized Tribonacci [32,33] Generalized Tetranacci [34,35] 2 AN APPLICATION OF THE…”

A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{3}$ and $\sum_{k=1}^{n}x^{k}W_{-k}^{3} $

“…Let x be a complex number. For n 0 we have the following formulas: If ( s 3 x 2 + rsx + 1)(r 3 x + s 3 x 2 + 3rsx 1) 6 = 0 then (a):…”

“…( s 3 x 2 + rsx + 1)(r3 x + s 3 x 2 + 3rsx 1)where 1 = x n+2 (s 3 x 2 + 2rsx 1)W 3 n+2 x n+1 (r 3 x + s 3 x 2 + 3r 2 s 2 x 2 r 3 s 3 x 3 + r 4 sx 2 + 2rsx 1)W 3 W 2 k W k+1 = 2 ( s 3 x 2 + rsx + 1)(r 3 x + s 3 x 2 + 3rsx 1) where 2 = rx n+2 (rsx 1) W 3 n+2 rs 3 x n+3 (rsx 1) W 3 n+1 + sx n+2 (2r 3 x s 3 x 2 + 1)W 2 n+2 Wn+1 x n+1 (r 3 x + s 3 x 2 + r 4 sx 2 2rs 4 x 3 + 2rsx 1)W 2 n+1 Wn+2 + rx (rsx 1) W 3 1 +rs 3 x 2 (rsx 1) W 3 0 sx(2r 3 x s 3 x 2 + 1)W 2 1 W0+(r 3 x + s 3 x 2 + r 4 sx 2 2rs 4 x…”

A Study on Generalized Tetranacci Numbers: Closed Form Formulas ∑n k=0 xkWk2 of Sums of the Squares of Terms

“…Name of sequence sums of second powers sums of third powers sums of powers Generalized Fibonacci [10,11,12,13,14,15,16,17,18] [19,20,21,22,23,24,25] [ 26,27,28,29] Generalized Tribonacci [30,31,32] Generalized Tetranacci [33,34,35] 2 An Application of the Sum of the Squares of the Numbers…”