Czechoslovak Mathematical Journal, Vol. 67, No. 4, pp. 919-936, 2017


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

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

Received April 3, 2016.   First published October 24, 2017.

Abstract:  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.
Keywords:  Riordan array; central coefficient; central Riordan array; generating function; Fuss-Catalan number; Pascal matrix; Catalan matrix
Classification MSC:  05A05, 05A10, 05A19, 15A09
DOI:  10.21136/CMJ.2017.0165-16

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Affiliations:   Sheng-Liang Yang, Yan-Xue Xu, Department of Applied Mathematics, Lanzhou University of Technology, Langongpink Street, Lanzhou, 730050, Gansu, P. R. China, e-mail: slyang@lut.cn, 903623012@qq.com; Tian-Xiao He, Department of Mathematics, Illinois Wesleyan University, 1312 Park Street, Bloomington, Illinois, 61702-2900, USA, e-mail: the@iwu.edu

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