On generalized Pell numbers of order $r\ge 2$

ABSTRACT

In this paper we investigate the generalized Pell numbers of order $r\ge 2$ through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order $r\ge 2$, are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order $r\ge 2$. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.

Keywords:
generalized Pell fundamental system; generalized Pell numbers; combinatorial identities; analytic representations; Generalized Cassini identity