On some identities for k-Fibonacci sequence

Abstract

We obtain some identities for k-Fibonacci numbers by using its Binet's formula. Also, another expression for the general term of the sequence, using the ordinary generating function, is provided.Mathematics Subject Classification: 11B37, 11B83, 05A15

A NOTE ON THE MODIFIED k-FIBONACCI-LIKE SEQUENCE

“…Binet's formulae are well known in the study of sequences like Fibonacci sequence [1,2,3,4,6,7,8,10,11,12]. In this section, we introduce and prove Binet's formula for the modified k-Fibonacci-like sequence.…”

“…Many authors who study sequences like Fibonacci sequence have introduced special identities, such as the Catalan, Cassini, and d'Ocagne identities [2,4,6,10,12]. They have then proved them using Binet's formula for each identity.…”

“…For any real number k, the k-Fibonacci sequence {F k,n } n∈N is defined by the recurrence relation (2) F k,n = kF k,n−1 + F k,n−2 for n ≥ 2 with F k,0 = 0 and F k,1 = 1. Edson and Yayenie [4] introduced the generalized Fibonacci sequence and proved some related identities.…”

Some properties of the generalized (p,q)- Fibonacci-Like number

“…Falcon and Plaza [1] showed some results of the k-Fibonacci sequence   k,n Then many researchers [3][4][5] showed some results of the k-Fibonacci-Like number in 2014.…”

On generalized K-Fibonacci sequence by two-cross-two matrix

“…Authors presented many interesting properties of k-Fibonacci numbers in [5,12]. In [13] authors defined k-Fibonacci numbers by using arithmetic indexes.…”