| Title:
|
$(m,r)$-central Riordan arrays and their applications (English) |
| Author:
|
Yang, Sheng-Liang |
| Author:
|
Xu, Yan-Xue |
| Author:
|
He, Tian-Xiao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
919-936 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For integers $m > r \geq 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d, h)=(d_{n,k})_{n,k \in \mathbb {N}}$ as $ d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots $, and the $(m,r)$-central coefficient triangle of $G$ as $$ G^{(m,r)} = (d_{mn+r,(m-1)n+k+r})_{n,k \in \mathbb {N}}. $$ It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h(0)=0$ and $d(0), h'(0)\not = 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$. Meanwhile, the algebraic structures of the $(m,r)$-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of $m$ and $r$. As applications, we determine the $(m,r)$-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach. (English) |
| Keyword:
|
Riordan array |
| Keyword:
|
central coefficient |
| Keyword:
|
central Riordan array |
| Keyword:
|
generating function |
| Keyword:
|
Fuss-Catalan number |
| Keyword:
|
Pascal matrix |
| Keyword:
|
Catalan matrix |
| MSC:
|
05A05 |
| MSC:
|
05A10 |
| MSC:
|
05A19 |
| MSC:
|
15A09 |
| idZBL:
|
Zbl 06819563 |
| idMR:
|
MR3736009 |
| DOI:
|
10.21136/CMJ.2017.0165-16 |
| . |
| Date available:
|
2017-11-20T14:52:33Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146957 |
| . |
| Reference:
|
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