Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.
Let denote by Fk,n the nth k-Fibonacci number where Fk,n = kFk,n−1+
Fk,n−2 for n 2 with initial conditions Fk,0 = 0, Fk,1 = 1, we may derive a functional
equation f(k, x) = kf(k, x − 1) + f(k, x − 2). In this paper, we solve this
equation and prove its Hyere-Ulam stability in the class of functions f : N×R ! X,
where X is a real Banach space.