Edmund Y. Chiang, Department of Mathematics, Baylor University

(on leave from The Hong Kong University of Science and Technology)

Discrete special functions: Heisenberg-Weyl algebra approach

We show that there is an holonomic PDEs (D-modules) approach to classical special functions. We show that both the classical special functions and their difference analogues, some of which have only been found recently, can be efficiently computed by the Heisenberg-Weyl algebraic framework. According to Truesdell, rudiments of algebraic approaches to special functions were already observed by some nineteenth century mathematicians. It only requires basic knowledge of solving linear PDEs. We illustrate our method with Bessel functions and the recently discovered difference Bessel functions in this talk. We shall touch upon this algebraic approach with a complex analytic and discrete Painleve equations connection.