Special Functions

In research, like many aspects of life, having the right tool for a problem is key. In mathematics and physics, the so-called special functions are often a crucial tool for solving many problems. You could say that for a function to be labelled as `special´, it needs to have many practical applications.

Special functions are a class of mathematical that arise frequently in various areas of mathematics and physics.  Special functions often appear as solutions of differential equations or integral equations, but many have also been constructed through purely algebraic or geometric constructions. Prominent examples of special functions include Euler's Gamma function, Gauss' hypergeometric function, and the Weierstrass elliptic function.

The theory of special functions is of great importance to mathematics and physics. It is driven by the discovery of sophisticated applications which require new functions with richer behaviour. The main purpose of this project's research is to advance the development of special functions required to solve the most general class of Calogero-Moser-Sutherland type and Ruijsenaars-van Diejen type models.

In the early 1970s it was discovered that quantum mechanical models for particles moving in only one spatial dimension and having a two-body interaction are exactly solvable given that their interaction is of a particular type. These types of exactly solvable models are nowadays called Calogero-Moser-Sutherland (CMS) type models. These CMS models have since contributed greatly to the development of physics and mathematics. Their contribution to mathematics has been particularly significant as CMS models appear in various fields, such as combinatorics, representation theory, harmonic analysis, random matrix theory, and many more. In particular, the CMS models have made significant contributions to the theory of special functions. This research area is highly interdisciplinary and researchers in various fields have had different motivations for studying these models, which has led to a fruitful development of the subject.

The CMS models often come in three kinds: rational, trigonometric/hyperbolic, and elliptic versions. In the trigonometric cases, the models have exact solutions given in terms of multivariate polynomials that generalize the classical orthogonal polynomials.