Some Properties of Convolved -Fibonacci Numbers

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. 1. Introduction Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]). The Fibonacci numbers are the terms of the sequence , wherein each term is the sum of the two previous terms, beginning with the values and . Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the -Fibonacci numbers. For any positive real number , the -Fibonacci sequence, say , is defined recurrently by In [2], -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 [2–7]. The convolved Fibonacci numbers？？ are defined by If we have classical Fibonacci numbers. Note that Moreover, using a result of Gould [8, page 699] on Humbert polynomials (with , and ), we have It seems that convolved Fibonacci numbers first appeared in the classical Riordan's book [9]. These numbers have been studied in several papers; see [10–12]. In this paper, we obtain new identities for convolved -Fibonacci numbers. 2. Some Properties of -Fibonacci Numbers and -Lucas Numbers The characteristic equation associated with the recurrence relation (1) is . The roots of this equation are Then we have the following basic identities: Some of the properties that the -Fibonacci numbers verify are summarized below (see [2, 6] for the proofs).(i)Binet formula: . (ii)Combinatorial？？formula: . (iii)Generating function: . Definition 1. For any positive real number , the -Lucas sequence, say , is defined recurrently by If？？ we have the classical Lucas numbers. Some properties that the -Lucas numbers verify are summarized below (see [13] for the proofs).(i)Binet formula: . (ii)Relation with -Fibonacci numbers: . 3. Convolved -Fibonacci Numbers Definition 2. The convolved -Fibonacci numbers？？ ？？are defined by Note that Moreover, from Humbert polynomials (with , and ), we have If we obtain the combinatorial formula of -Fibonacci numbers. In Tables 1, 2, and 3 some values of convolved -Fibonacci numbers are provided. The purpose of this paper is to investigate the properties of these numbers.