Costabile F.A. Modern Umbral Calculus: An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory

Boston: de Gruyter, 2019. — 266 p.The Umbral Calculus was described for the first time by John Blissard in the 1850’s in a form that we call “Classical”. After a short phase of success, the Umbral Calculus was largely rejected by the mathematics community due to the “lack of rigor”.
In the late 1960s the theory, worked out by Gian Carlo Rota and his co-workers, gave a completely rigorous formulation to the Umbral Calculus, which greatly rehabilitated it. The “Modern Classical” Umbral Calculus is now the systematic study of Sheffer polynomial sequences, including Binomial and Appell sequences. In fact, the Umbral Calculus, in Rota’s acceptation, allows an algebraic treatment of classical polynomials and numbers beginning from generating functions, recursive and reciprocity formulas, expansion theorems et cetera, depending on the choice of the formal power series as the “Umbra” (Latin for Shadow) of linear functionals and polynomials. Therefore, the Umbral Calculus is a mix of linear algebra, theory of formal power series, and classical analysis.
The Modern Umbral Calculus has been approached from different points of view. For example, by formal power series, algebraic, or operator
theoretic. Each of these approaches has been followed by many authors in different applications.
In recent times, it has been observed that there is an isomorphism between the Riordan matrices and the Sheffer polynomials ) (and hence also the Appell polynomials and Binomial sequences). At the same time, the possibility to define the Sheffer polynomials through determinantal forms
has been proved. Based on these dernier papers, in this book there is an attempt to present a theory of Modern Umbral Calculus in one variable, that is, known and also unknown results, using essentially elementary matrix calculus: lower triangular, infinite matrices, Hessemberg, Toeplitz, Riordan-type matrix, determinant and Cramer’s rule, recurrence relations, and few more. Hence, this book is not a complete and updated survey, but a new approach to the classic umbral calculus. In truth, the use of matrices in the theory of umbral calculus goes back to Vein’s papers