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Zeitschrift für Analysis und ihre Anwendungen

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Volume 20, Issue 4, 2001, pp. 1007–1029
DOI: 10.4171/ZAA/1057

Published online: 2001-12-31

On a New Type of Eisenstein Series in Clifford Analysis

Rolf Sören Kraußhar[1]

(1) Universiteit Gent, Belgium

In this paper we deduce a recursion formula for the partial derivatives of the fundamental solution of the generalized Cauchy-Riemann operator in $\mathbb R^{k+1}$ 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.

Keywords: Eisenstein series, Clifford analysis, Riemann zeta function, multiple divisor sums, permutational products

Kraußhar Rolf Sören: On a New Type of Eisenstein Series in Clifford Analysis. Z. Anal. Anwend. 20 (2001), 1007-1029. doi: 10.4171/ZAA/1057