Pascal eigenspaces and invariant sequences of the first or second kind

Abstract

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.

Riordan arrays, Chebyshev polynomials, Fibonacci bases

“…Eigenspaces of the transformationP M are subject of papers [7] - [9]. In [10], [11] eigenspaces of the transformations P M and P T M are considered on general terms; this point of view intersects with our observations set out in Section 4.…”

Inverse relations in Shapiro’s open questions

“…In fact, the row sums of F and DF −1 D are the Fibonacci and Catalan numbers, respectively. Among the objectives of our research is to present plausible answers to Q2, and also to give answers related to invariant sequences as self-inverse relations [8,9,15] to Q8 and Q8.1. More specifically, in 35 this paper, we investigate the structure of entries in DF −1 D and the role of DF −1 D in transforming invariant sequences, giving rise to answers for Q2.…”

“…More specifically, in 35 this paper, we investigate the structure of entries in DF −1 D and the role of DF −1 D in transforming invariant sequences, giving rise to answers for Q2. We also provide a method for constructing invariant sequences [8] by means of Riordan (pseudo) involutions, which allows us to answer both Q8 and Q8.1. 40 We begin with a slight extension of the notion of Riordan matrix in Definition 1.1.…”

“…F and L are P -invariant and inverse P -invariant sequences of the first kind, respectively, as a pseudo 55 involution P [5,8].…”

“…It can be easily proven that J(0)F and J(0)L are P -invariant and inverse P -invariant sequences of the second kind, respectively; see [8].…”