The Fourier expansion of the hypermonogenic generalized trigonometric and elliptic functions (Q1946716)

The authors consider multiperiodic paravector valued solutions of the Dirac-Hodge equation in the upper half-space of \(\mathbb R^n\). Such functions are usually called hypermonogenic. The corresponding Dirac-Hodge operator is a linearization of the Laplace-Beltrami operator. The Eisenstein series \[ \sum_{\omega\in\mathbb Z^n} (z+ 2\pi\omega)^{-1},\;m\geq n+1,\;z\in\mathbb R^{n+1},\;x_n> 0 \] are seen as \(n\)-fold hypermonogenic generalizations of the cotangent function. The main aim of the paper is the computation of Fourier series expansions of such Eisenstein series. The paper contains explicit formulae for their Fourier coefficients, which are constructed using Bell polynomials and Bessel functions.