(±1)-Invariant sequences and truncated Fibonacci sequences

Abstract 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 ∞  = {( x 0 ,  x 1 ,  x 2 , …) T ∣ x i ∈ 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 .