Certain matrices related to Fibonacci sequence having recursive entries

Abstract

Let φ = (φi)i≥1 and ψ = (ψi)i≥1 be two arbitrary sequences with φ1 = ψ1. LetAφ,ψ(n) denote the matrix of order n with entries ai,j , 1 ≤ i, j ≤ n, where a1,j = φj and ai,1 = ψi for 1 ≤ i ≤ n, and where ai,j = ai−1,j−1 + ai−1,j for 2 ≤ i, j ≤ n. It is of interest to evaluate the determinant of Aφ,ψ(n), where one of the sequences φ or ψ is the Fibonacci sequence (i.e., 1, 1, 2, 3, 5, 8,...) and the other is one of the following sequences: {k-times}α(k) = (1, 1,..., 1, 0, 0, 0,...) ,χ(k) = (1k, 2k, 3k,...,ik,...),ξ(k) = (1, k, k2,...,ki−1,...) (a geometric sequence),γ(k) = (1, 1 + k, 1+2k, ... , 1+(i − 1)k, .. .) (an arithmetic sequence). For some sequences of the above type the inverse of Aφ,ψ(n) is found. In the final part of this paper, the determinant of a generalized Pascal triangle associated to the Fibonacci sequence is found.