The Riordan group is the set of infinite lower triangular invertible matrices with the group operation given by a matrix multiplication that combines both the usual Cauchy product and the composition of formal power series. It is related to a broad family of polynomial sequences in one variable called Sheffer sequences. Riordan arrays and Sheffer sequences have various applications in Combinatorics, Analysis, Probability, etc.

In this talk I will present an enlightening symbolic treatment of the Riordan group and related Sheffer sequences based on a renewed approach to umbral calculus initiated by Gian Carlo Rota in the 90’s and further developed by Di Nardo and Senato in the first decade of the present century. The talk is based on joint work with Ângela Mestre, Pasquale Petrullo and Maria Manuel Torres.