(±1)-Invariant sequences and truncated Fibonacci sequences

Let P = [((i)(j))], (i, j = 0, 1, 2, . . .) and D=diag((- 1)(0), (-1)(1), (-1)(2), . . .). As a linear transformation of the infinite dimensional real vector space R-infinity = {(x(0), x(1), x(2), . . .)(T)\x(i) is an element of R for all i}, PD has only two eigenvalues 1, -1. In this paper, we find some matrices associated with P whose columns form bases for the eigenspaces for PD. We also introduce truncated Fibonacci sequences and truncated Lucas sequences and show that these sequences span the eigenspaces of PD. (C) 2004 Published by Elsevier Inc.