The Riordan group

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

“…A combination of these facts allows us to introduce Definition 2.1 The matrix (d n,k ) defined above is said to be a generalized Riordan array with respect to the basic set {p n (t)}. Following Shapiro-GetuWoan-Woodson [31], we write it by…”

“…They form a group, called the Riordan group (cf. 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.…”

Riordan matrices and sums of harmonic numbers

“…Riordan matrices form a group with respect to the matrix product [14], and this group admits several subgroups of combinatorial interest [16]. In particular, we have the derivative subgroup, consisting of the Riordan matrices (g(x), f (x)) such that g(x) = f ′ (x).…”

“…In enumerative combinatorics, Riordan matrices [14,15,16,5,7] form an important class of combinatorial objects. They are infinite lower triangular matrices R = [r n,k ] n,k 0 = (g(x), f (x)) whose columns have generating series r k (x) = g(x)f (x) k , where g(x) and f (x) are formal series with g 0 = 1, f 0 = 0 and f 1 = 0.…”

Hierarchical Dobi ski-type relations via substitution and the moment problem

“…Such matrices form a group, called the Riordan group, which has important applications in enumerative combinatorics [17], [18]. The purpose of this note is to place Eqs.…”