Zeitschrift für Analysis und ihre Anwendungen
Full-Text PDF (301 KB) | Metadata | Table of Contents | ZAA summary
Published online: 2002-09-30
Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and ApplicationsDenis Constales and Rolf Sören Kraußhar (1) Universiteit Gent, Belgium
(2) Universiteit Gent, Belgium
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 $\| q_n(z)\|$. 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 $q_n(z)$ 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.
Keywords: Cauchy-Riemann operator, fundamental solution, permutational products, hypercomplex variables, Dirichlet series
Constales Denis, Kraußhar Rolf Sören: Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications. Z. Anal. Anwend. 21 (2002), 579-597. doi: 10.4171/ZAA/1096