TY - JOUR

T1 - (m, r)-CENTRAL RIORDAN ARRAYS AND THEIR APPLICATIONS

AU - He, Tian-Xiao

AU - Yang, Sheng-liang

AU - Xu, Yan-Xue

N1 - For integers m r ≥ 0, Brietzke (2008) defined the ( m, r)-central coefficients of an infinite lower triangular matrix G = ( d, h) = ( d n, k) n, k∈N as d mn+ r,( m−1) n+ r, with n= 0, 1, 2,...,...

PY - 2017/10

Y1 - 2017/10

N2 - Excerpt from the abstract: 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) 6≠ 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.

AB - Excerpt from the abstract: 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) 6≠ 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.

KW - Riordan array

KW - central coefficient

KW - central Riordan array

KW - generating function

KW - Fuss-Catalan number

KW - Pascal matrix

KW - Catalan matrix

UR - https://link.springer.com/content/pdf/10.21136%2FCMJ.2017.0165-16.pdf

U2 - 10.21136/CMJ.2017.0165-16

DO - 10.21136/CMJ.2017.0165-16

M3 - Article

VL - 67

JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

ER -