Higher Order Teodorescu Operators and Cauchy-Pompeiu Type Formulas

Using the fundamental solution of the Helmholtz equation we construct the explicit form of the fundamental solution for powers of the factors of the Helmholtz operator in quaternionic analysis. These results lend assistance aid to investigate some properties of higher order Teodorescu operators. A fundamental solution can also be constructed for the product of Helmholtz operators. This is used to prove Cauchy–Pompeiu type representation formulas in quaternionic analysis for the general polynomial operator ∏j ν=1(D + αν) kν , where αν 6= αμ if ν 6= μ.