Moore–Penrose inverse of generalized Fibonacci matrix and its applications

Abstract

Let s be arbitrary integer, we introduce the notion of the matrix ${\mathcal{U}}_n^{(a,b,s)}$ of type s, whose nonzero entries are the classical Horadam numbers $U_n^{(a,b)}$. In this paper we consider singular case $s=-1$, then the Moore–Penrose inverse of the matrix ${\mathcal{U}}_n^{(a,b,-1)}$ is given. In the case $A=B=1$, we obtain the Pseudoinverse of the generalized Fibonacci matrix ${\mathcal{F}}_n^{(a,b,-1)}$. In addition, correlations between the matrix ${\mathcal{U}}_n^{(a,b,-1)}$ and the generalized Pascal matrices are discussed, and some combinatorial identities involving the Horadam numbers are derived.

Keywords:

• Moore–Penrose inverse
• Horadam number
• generalized Fibonacci number
• generalized Fibonacci matrix
• generalized Pascal matrix

2010 Mathematics Subject Classifications:

• 05A10
• 11B39
• 15A09

Disclosure

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Science Foundation of China [Grant No. 61272465], the Key Research Project of Higher school in Henan Province [Grant No. 15A110040].