On some generalizations of Horadam's numbers

Abstract

In this paper, we introduce the incomplete Horadam numbers $W_{n}(k)$, and hyper-Horadam numbers $W_{n}^{(k)}$, which generalize the Horadam's numbers defined by the recurrence $W_{n}=pW_{n-1}+qW_{n-2}$, with $W_{0}=a$ and $W_{1}=b$. We give some combinatorial properties. As an application, we evaluate a lower and upper bounds for the spectral norms of $r$-circulant matrices associated with these two generalizations. Moreover, we establish a new bounds for the spectral norms of $r$-circulant matrices associated with Horadam's numbers in terms of incomplete Horadam and hyper-Horadam numbers.