Edmund Y. Chiang, Baylor University (on leave from The Hong Kong University of Science and Technology)
Discrete special functions: a D-modules approach to special functions
We show that there is a holonomic D-modules (PDEs) 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 Weyl-algebraic framework. According to Truesdell, rudiments of algebraic approaches to special functions were already observed by some nineteenth century mathematicians. This algebraic method does not require Lie algebra theory explicitly apart from basic knowledge of solving linear PDEs. We illustrate our method with Bessel functions in this talk. We shall also explain the connection with this D-modules approach with recent advances in Nevanlinna theories for difference operators, which have their roots in the understanding of discrete Painleve equations.
