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.
An infinite real sequence {a n } is called an invariant sequence of the first (resp., second) kind if a n = n k=0. We review and investigate invariant sequences of the first and second kinds, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.
As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro's open questions [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596]. In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.
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