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$(m,r)$-central Riordan arrays and their applications
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 919-936 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/CMJ.2017.0165-16
Classification : 05A05, 05A10, 05A19, 15A09
Keywords: Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix 
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     title = {$(m,r)$-central {Riordan} arrays and their applications},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2017},
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Yang, Sheng-Liang; Xu, Yan-Xue; He, Tian-Xiao. $(m,r)$-central Riordan arrays and their applications. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 4, pp. 919-936. doi: 10.21136/CMJ.2017.0165-16

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