Generalized Riordan Arrays

The previous chapters have shown that the theory of Riordan arrays is a powerful tool for studying combinatorial sums and special polynomial and number sequences. One of the well-known classes of polynomial sequences is the class of Sheffer sequences, including many important sequences such as Bernoulli polynomials, Euler polynomials, Abel polynomials, Hermite polynomials, Laguerre polynomials, etc. This class contains the subclasses of associated sequences and Appell sequences. In [57–59], Rota, Roman, et al. established a solid background for Sheffer sequences by using the theory of modern umbral calculus and finite operator calculus. In [56], Roman further developed the theory of umbral calculus and generalized the concept of Sheffer sequences so that more special polynomial sequences are included such as the sequences of Gegenbauer polynomials, Chebyshev polynomials, and Jacobi polynomials. Using Roman’s notations, a generalized Sheffer sequence (sn(x))n∈N is defined by a generating function of the form 6.0.1 A(t)εx(B(t))=∑k=0∞sk(x)cktk, where εx(t)=∑k=0∞xktk/ck is a generalization of the exponential series, and (ck)k≥0 is a non-zero sequence with c0= 1. When ck= k!, (6.0.1) defines the (exponential) Sheffer sequence and its generating function turns into A(t) e xB(t). Note that there are several similar names presented in the literature, such as sequences of Sheffer A-type zero [63, 64] and generalized Appell sequences [9–12]. The connection between Riordan arrays and Sheffer sequences has already been pointed out by Shapiro et al. [62] and Sprugnoli [20, 65, 66]. In fact, the classical Riordan arrays are related to the 1-umbral calculus, and thus related to Sheffer sequences defined by (6.0.1) with ck= 1, and the exponential Riordan arrays shown in Sect. 6.1 are related to (exponential) Sheffer sequences with ck= k!. In other words, the exponential Riordan arrays are related to the k!-umbral calculus. In this chapter, we introduce the theory of exponential Riordan arrays and their production matrices in Sect. 6.1. Part of this section comes from Deutsch and Shapiro [26] and Barry [7, 8]. In Sects. 6.2 and 6.3, we introduce the concept of generalized Riordan arrays, and give explicitly the relationships between the generalized Riordan arrays and generalized Sheffer sequences. Then, we present the important properties and applications of the generalized Riordan arrays. Furthermore, the determinantal definition for Sheffer sequences using the relations between Riordan arrays and Sheffer sequences is also given. In Sect. 6.4, we introduce some important special Riordan arrays and Sheffer sequences. We will also give the basic properties of these arrays and polynomial sequences by using the results obtained in the previous two sections. Finally, in Sect. 6.5, we present the double Riordan arrays and their related Sheffer polynomial sequence pairs as well as their applications in combinatorics and series summations. Readers can also refer to the further discussion on the (exponential) Sheffer sequence and the classical Riordan array in He et al. [40] and the discussion on the generalized Sheffer sequence and the generalized Riordan array in Gould et al. [29], He [31–33], Wang [71], and Wang et al. [73].