arXiv:1103.5879v1 [math.CO] 30 Mar 2011 RIORDAN ARRAYS AND APPLICATIONS VIA THE CLASSICAL UMBRAL CALCULUS JOS ´ E AGAPITO, ˆ ANGELA MESTRE, PASQUALE PETRULLO, AND MARIA M. TORRES Abstract. We use the classical umbral calculus to describe Riordan arrays. Here, a Riordan array is generated by a pair of umbrae, and this provides efficient proofs of several basic results of the theory such as the multiplication rule, the recursive properties, the fundamental theorem and the connection with Sheffer sequences. In particular, we show that the fundamental theorem turns out to be a reformulation of the umbral Abel identity. As an application, we give an elementary approach to the problem of extending integer powers of Riordan arrays to complex powers in such a way that additivity of the exponents is preserved. Also, ordinary Riordan arrays are studied within the classical umbral perspective and some combinatorial identities are discussed regarding Catalan numbers, Fibonacci numbers and Chebyshev polynomials. 1. Introduction An exponential Riordan array is an infinite lower triangular matrix generated by two formal exponential series A(z )= ∑ n≥0 a n z n n! and B(z )= ∑ n≥1 b n z n n! , with a 0 = 0 and b 1 = 0, such that, denoting this matrix by (A, B), its (n, k)-entry is given by (A, B) n,k =  z n n!  A(z ) B(z ) k k!  . Riordan arrays were introduced by Shapiro et.al. in [26] as a generalization of the well known Pascal triangle and other types of combinatorial arrays (Rogers also studied in [20] equivalent generalizations of the Pascal triangle under the name of renewal arrays). Note that the generators A and B used in [26] are ordinary formal power series instead of series of exponential type. In this case the (n, k)-entry of (A, B) is given by (A, B) n,k =[z n ] ( A(z ) B(z ) k ) ). Riordan arrays form a group under matrix multiplication. They have been extensively studied and characterized in connection with combinatorial identities, recursion properties and walk problems (see for instance [5, 14, 25, 26, 28]). The purpose of this paper is to give a promising symbolic treatment of the exponential Riordan group based on a renewed approach to umbral calculus initiated in 1994 by Rota and Taylor [23]. In the last decade, this approach has been continued mostly by Di Nardo, Niederhausen and Senato [7, 10, 11] and more recently by Petrullo [9, 17, 19]. We will refer to this new version as the classical umbral calculus, to distinguish it from the more established treatment of umbral calculus using operator theory, as presented for instance in Roman’s book [21]. Under the old point of view of umbral calculus, Riordan arrays are also known as recursive matrices (see Barnabei, Brini and Nicoletti [2]). 2010 Mathematics Subject Classification. Primary 05A40, 05A15, 05A19, 11B83. This work was done within the activities of the Centro de Estruturas Lineares e Combinat´ orias (University of Lisbon, Portugal) and the Dipartamento di Matematica e Informatica (Universit` a degli Studi della Basilicata, Italy). The first author was partially supported by the Portuguese Science and Technology Foundation (FCT) through the program Ciˆ encia 2008 and grant PTDC/MAT/099880/2008; the second author was supported by the FCT grant SFRH/BPD/ 48223/2008 and the fourth author was partially supported by the FCT and FEDER/POCI 2010. 1