A & Z Sequences for Double Riordan Arrays

Abstract

AbstractA Riordan array is an infinite lower triangular matrix that is defined by two generating functions, g and f. The coefficients of the generating function g give the zeroth column and the nth column of the matrix is defined by the generating function \(gf^{n}\). We shall call f the multiplier function. Similarly, the Double Riordan array is an infinite lower triangular matrix that is defined by three generating functions, g, \(f_{1}\) and \(f_{2}\). Where the zeroth column of the Double Riordan array is g, the next column is given by \(gf_{1}\) and the following column will be defined by \(gf_{1}f_{2}\). The remaining columns are found by multiplying \(f_{1}\) and \(f_{2}\) alternatively. Thus, for a double Riordan array there are two multiplier functions, \(f_1\) and \(f_2.\) It is well known that any Riordan array can be determined by a Z-sequence and an A-sequence. This is the row construction of the array. This is not the case for Double Riordan arrays. In this paper, we show that double Riordan arrays can be determined by two Z-sequences and one A-sequence.KeywordsRiordan arrayDouble Riordan arrayA-sequenceZ-sequence