Inverse relations in Shapiro’s open questions

Abstract

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.

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.…”