Production Matrices and Riordan Arrays

Abstract

Abstract. We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of a production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method of Riordan matrices. Finally we treat the case of rational production matrices, i.e. those leading to rational generating functions.

Recurrence relations for the Sheffer sequences

“…Shapiro et al [13] introduced the concept of a Riordan array in 1991, then the concept is generalized to the exponential Riordan array by many authors [1,5,6,8,10,14,17]. The connection between the Riordan arrays and the Sheffer sequences has already been pointed out by Shapiro et al [13] and Sprugnoli [15,16].…”

“…Note that in this example a n (x) are the Bessel polynomials with exponents in decreasing order (see [5]). …”

“…(2) For example, the exponential Riordan array [1, e t − 1] corresponds to the Stirling matrix of the second kind, while the exponential Riordan array [1, log(1 + t)] is the Stirling matrix of the first kind. Lemma 2.1 [1,5,8]. Let A = (a n,k ) n,k 0 = [g(t), f (t)] be an exponential Riordan array and let…”

Riordan arrays and d-orthogonality

“…has exponential generating function e x . We will use the following [13,14], important result concerning matrices that are production matrices for exponential Riordan arrays.…”

A symbolic handling of Sheffer polynomials

“…In recent years, this subject has aroused some interest. We mention some results of recent literature that would benefit of the umbral approach, see [1,5,6,14,30]. If s n (x) = n k=0 s n,k x k is a polynomial of Sheffer type, then from (11) …”