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

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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 m n + r , ( m - 1 ) n + r , with n = 0 , 1 , 2 , ⋯ , and the ( m , r ) -central coefficient triangle of G as G ( m , r ) = ( d m n + r , ( m - 1 ) n + k + r ) n , k ∈ 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 ) ≠ 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...

A generalization of Pascal’s triangle using powers of base numbers

Gábor Kallós (2006)

Annales mathématiques Blaise Pascal

In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.

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