Abstract
We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.
Highlights
Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art
If r = 1 we obtain the combinatorial formula of k-Fibonacci numbers
In the theorem we show that the convolved kFibonacci numbers can be expressed in terms of k-Fibonacci and k-Lucas numbers
Summary
Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]). In [2], k-Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. These numbers have been studied in several papers; see [2,3,4,5,6,7]. Using a result of Gould [8, page 699] on Humbert polynomials (with n = j, m = 2, x = 1/2, y = −1, p = −r, and C = 1), we have It seems that convolved Fibonacci numbers first appeared in the classical Riordan’s book [9]. We obtain new identities for convolved k-Fibonacci numbers
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