Simple proofs of open problems about the structure of involutions in the Riordan group

Shift operators defined in the Riordan group and their applications

“…In [2], an element of order 2 in the Riordan group is called a Riordan involution. Some structures of a Riordan involution were presented in [1,2]. Proposition 2.7.…”

“…It is obvious that the key step to find Riordan involution is to solve the Babbage equation h(h(t)) = t. [1,13] gave a nice method in solving this equation. Since h(t) ∈ F 1 and h(t) = n≥0 h n t n , a direct computation of h(t) using Faá di Bruno's formula might be presented below.…”

“…, is the minimal number r ∈ N such that f r = 0; F r is the set of formal power series of order r. Let d(t) ∈ F 0 and h(t) ∈ F 1 ; the pair (d(t), h(t)) defines the (proper) Riordan array D = (d n,k ) n,k∈N = (d(t), h(t)) having (1) or, in other words, having d(t)h(t) k as the generating function whose coefficients make-up the entries of column k.…”

Pascal eigenspaces and invariant sequences of the first or second kind

“…Invariant sequences, which are also called self-inverse sequences in [15], have indeed been studied by several authors [6,8,14,15]. They are naturally connected to involutory (also known as involution or self-invertible) matrices [9] and to Riordan involutions [5]. Involutory matrices find use in numerical methods for differential equations [2,9].…”

“…They are also useful in cryptography, information theory, and computer security by 15 providing convenient encryption and decryption methods [1]. Motivated by Shapiro's open questions [12], Riordan involutions have been intensely investigated as a combinatorial concept [4,5]. In this paper, we investigate invariant sequences by means of the eigenspaces of P D and P T D, where P is the Pascal matrix and D an infinite diagonal matrix with alternating diagonal entries in {1, −1} (see Sections 2,3).…”