On paravector valued homogeneous monogenic polynomials with binomial expansion
Starting from a set of paravector valued homogeneous monogenic polynomials the authors construct a sequence of generalized Appell polynomials. Two different types of polynomials are studied. One of them is given by (analogous to the complex powers \(z^k\)) \[ {\mathcal P}^k(x_0,\widehat x)= (x_0+ (i_1 x_1+ i_2x_2)(i_1 e_1+ i_2 e_2))^k. \] This generation of polynomials is an alternative to the Fueter-Sce construction (only for odd integer dimension) and the Cauchy-Kovalevskaya extension. The proofs are lined out in the case \(n=2\). An important role plays the constant imaginary vector \[ \widehat i= (i_1 e_1+\cdots+ i_n e_n) \] with \(\widehat i^2=-1\), i.e., \(\widehat i\) is a generalized unit vector.
scientific article; zbMATH DE number 6123928
scientific article