Incomplete -Fibonacci and -Lucas Numbers

We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers. 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]). Fibonacci numbers are defined by the recurrence relation There exist a lot of properties about Fibonacci numbers. In particular, there is a beautiful combinatorial identity to Fibonacci numbers [1] From (2), Filipponi [2] introduced the incomplete Fibonacci numbers and the incomplete Lucas numbers . They are defined by Further in [3], generating functions of the incomplete Fibonacci and Lucas numbers are determined. In [4], Djordjevi？ gave the incomplete generalized Fibonacci and Lucas numbers. In [5], Djordjevi？ and Srivastava defined incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers. In [6], the authors define the incomplete Fibonacci and Lucas -numbers. Also the authors define the incomplete bivariate Fibonacci and Lucas -polynomials in [7]. On the other hand, many kinds of generalizations of Fibonacci 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 [8], -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 [8–14]. For any positive real number , the -Lucas sequence, say , is defined recurrently by If , we have the classical Lucas numbers. Moreover, ; see [15]. In [12], the explicit formula to -Fibonacci numbers is and the explicit formula of -Lucas numbers is From (6) and (7), we introduce the incomplete -Fibonacci and -Lucas numbers and we obtain new recurrent relations, new identities, and their generating functions. 2. The Incomplete -Fibonacci Numbers Definition 1. The incomplete -Fibonacci numbers are defined by In Table 1, some values of incomplete -Fibonacci numbers are provided. Table 1: The numbers , for . We note that For , we get incomplete Fibonacci numbers [2]. Some special cases of (8) are 2.1. Some Recurrence Properties of the Numbers Proposition 2. The recurrence relation of the incomplete -Fibonacci numbers is The relation (11) can be transformed into the nonhomogeneous recurrence relation Proof. Use Definition 1 to rewrite the right-hand side of (11) as Proposition 3. One has Proof (by induction on ). Sum (14) clearly holds