We develop a D-module approach to generating functions for Bessel functions in this thesis. By solving a specific system of holonomic PDEs of Bessel modules from the viewpoint of Bernstein's holonomic module theory, we show that the difference Bessel equations and its solutions recently discovered by Bohner and Cuchta and the classical Bessel functions can be unified and derived under this Weyl-algebraic interpretation. We have discovered difference Bessel polynomials from studying the Bessel polynomial module. This Bessel polynomial module allows us to derive the delay-difference formulae and generating functions for the difference Bessel polynomials and classical Bessel polynomials. Then we study the orthogonality of these Bessel polynomials as a residue map which is a left D-linear m...[ Read more ]

We develop a D-module approach to generating functions for Bessel functions in this thesis. By solving a specific system of holonomic PDEs of Bessel modules from the viewpoint of Bernstein's holonomic module theory, we show that the difference Bessel equations and its solutions recently discovered by Bohner and Cuchta and the classical Bessel functions can be unified and derived under this Weyl-algebraic interpretation. We have discovered difference Bessel polynomials from studying the Bessel polynomial module. This Bessel polynomial module allows us to derive the delay-difference formulae and generating functions for the difference Bessel polynomials and classical Bessel polynomials. Then we study the orthogonality of these Bessel polynomials as a residue map which is a left D-linear map. In the case of difference Bessel polynomials, the residue map is given by a Barnes integral. Similarly, we construct a q-Bessel module base on a q-deformation of the commutator set up to study the generating functions of Jackson's q-Bessel functions. Finally, we also use a fractional commutator to study and unify different notions of fractional derivatives and fractional differences found in the literature.