Basic set of polynomial solutions for the iterated vector-matrix Lamé equation (Q2772153)

Construction and investigation of a basic set of polynomial solutions to the iterated vector-matrix Lamé equation NEWLINE\[NEWLINE \left(\begin{pmatrix}\Delta&0&0\\0&\Delta&0\\0&0&\Delta \end{pmatrix}+ \frac 1{1-2\nu}\begin{pmatrix}\partial_{xx}& \partial_{xy}&\partial_{xz}\\ \partial_{yx}&\partial_{yy}& \partial_{yz}\\ \partial_{zx}&\partial_{zy}&\partial_{zz} \end{pmatrix}\right)^pu_p(x, y, z)=0, \tag{*} NEWLINE\]NEWLINE where \(\nu\) is the Poisson coefficient, on the basis of a set of polyharmonic polynomials normalized with respect to the Laplace operator \(\Delta\) are considered. It is proved that the basic set of polynomial solutions of degree \(n\) to the equation (*) consists of \(3p(2n-2p+3)\) polynomials.