On rotationally symmetric Dirac equations and hypergeometric functions I

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 \(D := \sum^n_{i=1} \frac{\partial}{{\partial}x_i}e_i\) be the Euclidean Dirac operator in the n-dimensional flat space \({{\mathbb{R}}}^n, E := \sum^n_{i=1}x_i\frac{\partial}{{\partial}x_i}\) the radial symmetric Euler operator and α and λ 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 2 F 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 λ → 0.