The generalized Fibonomial matrix

Abstract

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k + 1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples. (C) 2009 Elsevier Ltd. All rights reserved.