d−Fibonacci and d−Lucas polynomials

The fibonacci numbers fn are defined by the recurrence sequence fn = fn−1 + fn−2, for all n ≥ 1 with the initial conditions f0 = 0 and f1 = 1. this famous sequence appears in many areas of mathematics and it has been generalized in many research fields. we recall here the generalization of falcon and plaza [4], where a general fibonacci sequence was introduced. it generalizes, among others, both the classical fibonacci sequence and pell sequence. these general k−fibonacci numbers fk,n are defined by fk,n = kfk,n−1 + fk,n−2, n ≥ 2, with the initial values f0 = 0 and f1 = 1. we note that the pell numbers are 2−fibonacci numbers. it should be reminded that the k−fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge partition. besides that, the k−fibonacci numbers were given in an explicit way and many properties were proven in [12]. they were related with the so-called pascal 2−triangle. in fact, the fibonacci polynomials are defined by using the fibonacci-like recursion relations. they were studied in 1883 by the belgian mathematician catalan and jacobsthal. in what follows we