Archiv der Mathematik vol:90 issue:5 pages:440-449

Abstract:

In this paper we establish an interesting relationship between the classical hypergeometric functions and solutions to a special class of radial symmetric higher dimensional Dirac type equations and describe how these equations can be solved fully analytically with methods from hypercomplex analysis. Concretely, let $D :=\sum_{i=1}^n \frac{\partial }{\partial x_i} e_i$ be the
Euclidean Dirac operator in the $n$-dimensional flat space
$\mathbb{R}^{n}$,
$E:=\sum_{i=1}^n x_i \frac{\partial }{\partial x_i}$ the radial symmetric Euler operator and $\alpha$ and
$\lambda$ be arbitrary non-zero complex parameters. We set up an explicit description of the Clifford algebra valued solutions to the PDE system $[D - \lambda - \alpha {\bf x}E]f({\bf x}) = 0$ ($
{\bf x} \in \Omega \subseteq \mathbb{R}^n)$ in terms of
hypergeometric functions ${}_2F_1(a,b;c;z)$ of arbitrary complex parameters $a,b$ and half-integer parameter $c$ and special homogeneous polynomials. The regular solutions to the Dirac equation on the real projective space $\mathbb{R}^{1,n}$ which recently attracted much interest are recovered in the limit case $\lambda \to 0$.