$(m,r)$-central Riordan arrays and their applications

Czechoslovak Mathematical Journal

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He¹ 

关键词: Riordan array;    central coefficient;    central Riordan array;    generating function;    Fuss-Catalan number;    Pascal matrix;    Catalan matrix;   

DOI  :  10.21136/CMJ.2017.0165-16

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

  For integers $m > r \geq0$, 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.

【 授权许可】

Unknown