On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

Abstract.

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations.

Let $D: = {\sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_{i} }}e_{i}}}$ be the Euclidean Dirac operator in the n-dimensional flat space $\mathbb{R}^{n},\;{\mathbf{E}}: = {\sum\limits_{i = 1}^n {x_{i} \frac{\partial }{{\partial x_{i} }}}}$ the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system $\left[ {D - \lambda - (1 + \alpha )\frac{{\text{x}}} {{\text{|x|}}^{2}}{\mathbf{E}}} \right]f = 0$ in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.