Inversion of the generalized Fibonacci matrix by convolution

Abstract

A convolution formula containing the generalized Fibonacci numbers and applications of this formula are investigated. Starting from the convolution formula, we derive combinatorial identities involving generalized and usual Fibonacci numbers, as well as the Lucas numbers. The inversion of a lower triangular matrix and the generalized inversion of strictly lower triangular Toeplitz matrix whose non-zero elements are generalized Fibonacci numbers are considered.

Keywords:

• generalized Fibonacci numbers
• generalized Fibonacci matrix
• convolution formula
• matrix inversion
• Moore–Penrose inverse

2010 AMS Subject Classifications :

• 15A09
• 11B39
• 05A19