Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914971722448896 |
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| author | Frankson, Shakuan |
| author_facet | Frankson, Shakuan |
| contents | The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, $g$ and $f$. The elements of the group are called Riordan arrays, denoted by $(g,f)$, and the $k$th column of a Riordan array is given by the function $gf^k$. The Double Riordan group is defined similarly using three generating functions $g$, $f_1$, and $f_2$, where $g$ is an even function and $f_1$ and $f_2$ are odd functions. This group generalizes the Checkerboard subgroup of the Riordan group, where $g$ is even and $f$ is odd. An open question posed by Davenport, Shapiro, and Woodson was if there exists an isomorphism between the Riordan group and a subgroup of the Double Riordan group. This question is answered in this article. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group Frankson, Shakuan Combinatorics Group Theory The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, $g$ and $f$. The elements of the group are called Riordan arrays, denoted by $(g,f)$, and the $k$th column of a Riordan array is given by the function $gf^k$. The Double Riordan group is defined similarly using three generating functions $g$, $f_1$, and $f_2$, where $g$ is an even function and $f_1$ and $f_2$ are odd functions. This group generalizes the Checkerboard subgroup of the Riordan group, where $g$ is even and $f$ is odd. An open question posed by Davenport, Shapiro, and Woodson was if there exists an isomorphism between the Riordan group and a subgroup of the Double Riordan group. This question is answered in this article. |
| title | Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2410.10709 |