Umbral Calculus:

Lecturer Contact Information:

Email: e.lytvynov@swansea.ac.uk

Description

Umbral calculus (also called calculus of finite differences) is essentially the theory of Sheffer polynomial sequences, which are characterised by the exponential form of their generating function. The class of Sheffer sequences includes the binomial sequences and  Appell sequences.  After a long period when one-dimensional umbral calculus was used for purely formal calculations, the theory became rigorous in the 1970s due to the seminal works of Gian-Carlo Rota, Steven Roman and their co-authors. Their theory is nowadays called the modern umbral calculus. Umbral calculus found applications in combinatorics, theory of special functions, approximation theory, probability and statistics, algebra, topology and physics.

The course will discuss the following topics:

1.     Introduction to modern umbral calculus;

2.     The group structure of the set of Sheffer polynomials (umbral group, Sheffer group);

3.     Sheffer group as a Lie group and its Lie algebra;

4.     Spaces of entire functions and their description through an expansion of an entire function in Sheffer polynomials.

Pre-requisites: complex analysis at an undergraduate level. Some familiarity with matrix Lie groups and Lie algebras would be of advantage.

Lectures are on Fridays, 13:00-15:00, starting on 28 April for 8 weeks.