Sums of the squares of terms of sequence {u n }

Abstract

In this paper, we consider generalized Fibonacci type second order linear recurrence {u n }. We derive a generating matrix for both the sums of squares, n i=0 u 2 i and the products of the form u n u n+2 . We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms u n u n+1 as well as the product, u n u n+2 . Further we give generating functions and combinatorial representations of the sums of squares of terms of {u n } and the product, u n u n+2 .

Cholesky Factorization of the Generalized Symmetric k- Fibonacci Matrix

“…4. [10] Let 𝐹 𝑘,𝑛 be the 𝑛 −th term of the sequence {𝐹 𝑘,𝑛 } 𝑛∈ℕ ∞ . Then we have Therefore the proof is complete.…”

Generating functions of binary products of k-Fibonacci and orthogonal polynomials

“…In this part, we now derive the new generating functions of the products of some known numbers. For the applications of generating functions of some known functions, we refer the reader to see the references [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. For the case A = {a 1 , −a 2 } and E = {e 1 , −e 2 } with replacing a 2 by −a 2 , e 2 by −e 2 in (3.1), we have…”

Sums of powers of Fibonacci polynomials

“…For d = 2, these sums have been evaluated in [1] using matrix methods, which resulted in some quite long calculations. We decided to replace the parameter A from [1] by x to emphasize the nature of (Fibonacci) polynomials.…”