New Generalizations of Lucas Numbers

In this paper, we present new generalizations of the Lucas numbers by matrix representation, using Generalized Lucas Polynomials. These new generalizations include more powerful relationships with generalizations of Fibonacci numbers. We give some properties of these new generalizations and obtain some relations between the generalized order-k Lucas numbers and the generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.