Riordan arrays and the Abel-Gould identity

Riordan Matrix Representations of Euler’s Constant γ and Euler’s Number e

“…Standard references for Riordan matrices are Shapiro et al [3] and Sprugnoli [4,5]. The notation ( , ) employed above is not the typical one, so we explain the connection.…”

“…Our ultimate goal in this section is to explain how Riordan matrices are connected to a permutation representation : → ( ) of a certain group acting on an infinitedimensional vector space . Some of the notation in the sequel will differ from standard notation such as that given by Shapiro et al [3] and Sprugnoli [4,5], but we will explain the connection.…”

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

“…Inspired by [7,33], we will use the fundamental property of the generalized Riordan arrays and its alternating form with respect to basic sets to give two ways for the construction of a general class of Abel identities. First, we present the fundamental property of the generalized Riordan arrays with respect to a basic set using a similar argument of [28].…”

“…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.…”

Inverse Relations and Schauder Bases

“…Besides inverse relations, our applications of Schauder bases include an Abel identity, a Gould identity, their analogues and a generalization. The Abel and Gould identities were also generalized by Sprugnoli [27] using Riordan arrays. We will examine the concept of Riordan arrays using Schauder bases.…”