On the representation of k sequences of generalized order-k numbers

Abstract: In this paper, we define k-generalized order-k numbers and we obtain a relation between i-th sequences and k-th sequences of k-generalized order-k numbers. We give some determinantal and permanental representations of k-generalized order-k numbers by using various matrices. Using the relation between i-th sequences and k-th sequences of k-generalized order-k numbers we give some determinantal and permanental representations of i-th sequences of generalized order-k numbers. In addition, we obtain Binet's formula for generalized order-k Pell numbers by using our representations.