Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications

Abstract

In this paper, we discuss several essentially different formulas for the general derivatives $q_n(z)$ of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis, upon which – among other important applications – the theory of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition methods, we obtain a compact integral representation formula over a half-space, which also lends itself to establish upper bounds on the values $\Vert q_n(z)\Vert$. A second formula that we discuss is a recurrence formula involving permutational products of hypercomplex variables by which these estimates can be obtained immediately. We further prove several formulas for $q_n(z)$ in terms of explicit, non-recurrent finite sums, leading themselves to further representations in terms of permutational products but using different and fewer hypercomplex variables than used in the recurrence relations. Summing up a fixed $q_n$ over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for $q_n(z)$ to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.