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

For integers $m>r\ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d,h)={\left({d}_{n,k}\right)}_{n,k\in \mathbb{N}}$ as ${d}_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots $, and the $(m,r)$-central coefficient triangle of $G$ as $${G}^{(m,r)}={\left({d}_{mn+r,(m-1)n+k+r}\right)}_{n,k\in \mathbb{N}}.$$ 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 $h\left(0\right)=0$ and $d\left(0\right),{h}^{\text{'}}\left(0\right)\ne 0$, 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...