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Keywords:
Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix
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.
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