Extensions of the Riordan Group
- Authors
- Shapiro, L.; Sprugnoli, R.; Barry, P.; Cheon, G.-S.; He, T.-X.; Merlini, D.; Wang, W.
- Issue Date
- 2022
- Publisher
- Springer Science and Business Media Deutschland GmbH
- Citation
- Springer Monographs in Mathematics, pp 213 - 241
- Pages
- 29
- Indexed
- SCOPUS
- Journal Title
- Springer Monographs in Mathematics
- Start Page
- 213
- End Page
- 241
- URI
- https://scholarworks.bwise.kr/skku/handle/2021.sw.skku/98856
- DOI
- 10.1007/978-3-030-94151-2_7
- ISSN
- 1439-7382
- Abstract
- We consider two groups, F0= { g∈ F[ [ z] ] ∣ g(0 ) ≠ 0 } under multiplication, and F1= zF0 under composition where F is the real field R or complex field C. As observed in Sect. 3.3, it is known that the Riordan group R is isomorphic to the semidirect product F0⋊ F1. It may be viewed as a group extension of F1 by F0. In this chapter, we develop the group of three-dimensional Riordan arrays [7] from an extension of the Riordan group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. Moreover, we discuss the multivariate Riordan group [6, 17] defined by the semidirect product F0d⋊F1d for an integer d≥ 1. The two groups F0d and F1d are obtained, respectively, from F0 and F1 by extending the ring F[ [ z] ] of a single variable to the ring F[ [ z1, …, zd] ] of d variables. We will see some similarity with the Riordan group in a single variable, but its matrix representation, called a multivariate Riordan array, will be quite different to usual Riordan arrays. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
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