On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays
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On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays Paul Barry1 · Nikolaos Pantelidis1 Received: 1 August 2020 / Accepted: 6 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The group of almost Riordan arrays contains the group of Riordan arrays as a subgroup. In this note, we exhibit examples of pseudo-involutions, conditions under which we define an involution, and methods of constructing quasi-involutions in the group of almost Riordan arrays. Keywords Riordan arrays · Almost Riordan arrays · Involution · Pseudo-involution · Quasi-involution · Quasi-transitional matrix · Quasi-compression
1 Introduction Since the initial paper [14] defining the Riordan group, there has been interest in studying involutions, pseudo-involutions and quasi-involutions [6–8,10] associated to this group. The idea of Riordan matrices with an extra column has origins from the expression of the L DU decomposition for certain symmetric Toeplitz-plus-Hankel matrix [1]. In this note, we take a look at involutions, pseudo-involutions and quasiinvolutions associated to the related group of almost Riordan arrays [2]. In this first section, we recall the definition of the Riordan group, and the definition of the group of almost Riordan arrays (of order 1, initially). We then proceed to look at almost Riordan pseudo-involutions of order one and two, and to conditions that allow us to define involutions in the almost Riordan group. In the last section, we present two methods of building a form of quasi-involutions of the almost Riordan group.
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Nikolaos Pantelidis [email protected] Paul Barry [email protected]
1
Department of Computing and Mathematics, Waterford Institute of Technology, Cork Road, Waterford, Ireland
123
Journal of Algebraic Combinatorics
We define Fr to be the set of formal power series of order r , where Fr = {ar x r + ar +1 x r +1 + ar +2 x r +2 + · · · |ai ∈ R} where R is a suitable ring with unit (which we shall denote by 1). In the sequel, we shall take R = Z. The Riordan group over R is then given by the semi-direct product R = F0 F 1 . To an element g(x), f (x) ∈ R we associate the R-matrix with (n, k)-th element Tn,k = [x n ]g(x) f (x)k . This is an invertible lower-triangular matrix. For g(x) ∈ F0 and f (x) ∈ F1 , we shall also use the notation g(x), f (x) to represent the matrix that begins ⎛
⎞ g0 0 0 0 ⎜ g1 g0 f 1 0 0 ⎟ ⎜ ⎟. 2 ⎝ g2 g0 f 2 + g1 f 1 g0 f 1 0 ⎠ g3 g0 f 3 + g1 f 2 + g2 f 1 2g0 f 1 f 2 + g1 f 12 g0 f 13 Example 1 The Riordan array
matrix (Pascal’s triangle) B =
1 x , 1−x
1−x n k . We
has associated matrix equal to the binomial have
xk 1 1 − x (1 − x)k 1 = [x n−k ] (1 − x)k+1
−(k + 1) = [x n−k ] (−1) j x j j
k+1+ j −1 j = [x n−k ] x j
k+n−k = n−k n = , k
Tn,k = [x n ]
123
Journal of Algebraic Combinatorics
and we write ⎛
1 ⎜1
⎜ ⎜1 1 x ⎜ = ⎜1 , ⎜ 1−x 1−x ⎜1 ⎝ .. .
0 1 2 3 4 .. .
0 0 1 3 6 .. .
0 0 0 1 4 .. .
0 0 0 0 1 .. .
⎞ ··· ···⎟ ⎟ ···⎟ ⎟ . ···⎟ ⎟ ⎟ ···⎠ ..
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