Title: Representation formulas for the general derivatives of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis and Applications Authors: Constales, Denis ×Krausshar, Rolf Sören # Issue Date: 2002 Publisher: Heldermann Verlag Series Title: Zeitschrift für Analysis und ihre Anwendungen vol:21 issue:3 pages:579-597 Abstract: In this paper, we discuss several essentially different formulas for the general derivatives $q_{\bf n}(z)$ 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 $\|q_{\bf n}(z)\|$. 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 $q_{\bf n}(z)$ 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 $q_{\bf n}$ over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for $q_{\bf n}(z)$ to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series. ISSN: 0232-2064 Publication status: published KU Leuven publication type: IT Appears in Collections: Analysis Section