Production Matrices of Double Riordan Arrays

A double Riordan array is an infinite lower triangular matrix, denoted by \( (g; f_1, f_2)\) , where g, \(f_1\) , and \(f_2\) are generating functions. The coefficients of the generating function g gives the first column of the matrix, and the remaining columns are found by multiplying the previous column by alternating \(f_1\) and \(f_2\) . In other words, \(\displaystyle (g; f_1, f_2)=(g, gf_1, gf_1f_2,g{f_1}^2f_2, gf_1^2f_2^2,\dots ). \) This is the columns construction of a double Riordan array. We can determine the elements of a double Riordan array using A- and Z-sequences which gives a row construction of a double Riordan array, see ([2] and [5]). In this chapter we define the production matrix of a double Riordan array, and show how it can be used to determine the A- and Z-sequences.