In this paper we deduce a recursion formula for the partial derivatives of the fundamental solution of the generalized Cauchy-Riemann operator in in terms of permutational products. These functions generalize the classical negative power functions to Clifford analysis. We exploit them to introduce a new generalization of the classical complex analytic Eisenstein series on the half-plane to higher dimensions satisfying the generalized Cauchy-Riemann differential equation. Under function-theoretical and number-theoretical aspects we investigate their Fourier series expansion in which multiple divisor sums and certain generalizations of the Riemann zeta function play a crucial role.
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Rolf Sören Kraußhar, On a New Type of Eisenstein Series in Clifford Analysis. Z. Anal. Anwend. 20 (2001), no. 4, pp. 1007–1029DOI 10.4171/ZAA/1057