On Some Alternative Characterizations of Riordan Arrays

Abstract

ABSTRACT. We give several new characterizations of Riordan Arrays, the most important of which is: if fd nÒk g nÒk2N is a lower triangular array whose generic element d nÒk linearly depends on the elements in a well-defined though large area of the array, then fd nÒk g nÒk2N is Riordan. We also provide some applications of these characterizations to the lattice path theory.

Combinatorial sums through Riordan arrays

“…There exists a large literature on Riordan arrays; in chronological order, we can quote here the papers [4,5,8,11,13,[15][16][17][18][19].…”

A refined enumeration of hex trees and related polynomials

“…The identity is (1, z), and the inverse of (g, f ) is (1/(g •f ),f ) wheref is the compositional inverse of f . It is known [10] that a Riordan matrix (g, f ) = [r n,k ] is associated to two sequences A = (a n ) n≥0 and Z = (z n ) n≥0 with a 0 = 0 such that r n+1,k+1 = j≥0 a j r n,k+j and r n+1,0 = j≥0 z j r n,j , or equivalently…”

On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

“…[13] presents the isomorphism between the Riordan group and Sheffer group. Rogers [24] introduced the concept of the A-sequence for Riordan arrays; Merlini, Rogers, Sprugnoli, and Verri [20] introduced the related concept of the Z-sequence and showed that these two concepts, together with the element d 0,0 , completely characterize a proper Riordan array. He and Sprugnoli [16] presented the characterization of Riordan arrays by means of the A-and Z-sequences for some subgroups of R and the products and the inverses of Riordan arrays.…”

“…Shapiro, Getu, Woan, and Woodson [31]). Some of the main results on the Riordan group and its application to combinatorial sums and identities can be found in Sprugnoli [32,33], on subgroups of the Riordan group in Peart and Woan [23] and Shapiro [28], on some characterizations of Riordan matrices in Rogers [24], Merlini, Rogers, Sprugnoli, and Verri [20], and He and Sprugnoli [16], and on many interesting related results in Cheon, Kim, and Shapiro [2,3], He [9], He, Hsu, and Shiue [13], Nkwanta [22], Shapiro [29,30], Wang and Wang [34], Yang, Zheng, Yuan, and He [36] , and so forth.…”