On convolved generalized Fibonacci and Lucas polynomials

We define the convolved hðxÞ-Fibonacci polynomials as an extension of the classical con-volved Fibonacci numbers. Then we give some combinatorial formulas involving the hðxÞ-Fibonacci and hðxÞ-Lucas polynomials. Moreover we obtain the convolved hðxÞ-Fibo-nacci polynomials from a family of Hessenberg matrices. Fibonacci numbers and their generalizations have many interesting properties and applications to almost every fields of science and art (e.g., see [1]). The Fibonacci numbers F n are the terms of the sequence 0; 1; 1; 2; 3; 5;. . ., wherein each term is the sum of the two previous terms, beginning with the values F 0 ¼ 0 and F 1 ¼ 1. Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the k-Fibonacci Numbers. 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 [2–7].