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Fundamentals of Numerical Analysis and Symbolic Computation

Lecturer: Prof. Diego Dominici

Lecture no.: 326.004

ECTS: 3, hours: 2

Time: Wed, 11:00-12:30, SP 416-1

No lecture on March 27 and May 15.


An introduction to the Umbral calculus

The Umbral calculus was developed in the 1800s and is attributed to various combinations of John Blissard, Édouard Lucas, and James Joseph Sylvester. The early days of umbral calculus were like the early days of the infinitesimal calculus: in the hands of skilled practitioners it produced correct results, but no one was sure why it worked. E. T. Bell was fascinated by the umbral calculus and attempted a revival of it in the 1930s and 1940s, but still no one understood it well. Gian-Carlo Rota became interested in it in the 1960s and 1970s and put a more general form of umbral calculus on a rigorous basis. The term umbral was introduced by Sylvester from "umbra", Latin for shadow.

In this course, we will present some of the basic ideas of the umbral calculus, formalized in terms of linear functionals on the space of polynomials. As usual, the functionals form a linear space (the dual space), but we will turn this into an algebra by defining a multiplication corresponding to the Cauchy product of coefficients that results from multiplying two power series, so the product of two functionals matches the product of the corresponding series. We will then develop several standard operators (functionals) and their properties.

Some topics to be covered will include:  Sheffer sequences, finite differences, generating functions, Newton series, special functions, and harmonic numbers.

No special background will be required, so everyone is welcome.