The Relativistic Hermite Polynomials and the Wave Equation

Solutions of the homogeneous 2D scalar wave equation of a type reminiscent of the “splash pulse” waveform are investigated in some detail. In particular, it is shown that the “higher-order” solutions relative to a given “fundamental” one, from which they are obtained through a definite “generation scheme”, come to involve the relativistic Hermite polynomials. This parallels the results of a previous work, where solutions of the 3D wave equation involving the relativistic Laguerre polynomials have been suggested. Then, exploiting a well known rule, the obtained wave functions are used to construct further solutions of the 3D wave equation. The link of the resulting wave functions with those analyzed in the previous work is clarified, the pertinent generation scheme being indeed inferred. Finally, solutions of the Klein-Gordon equation which relate to such Lorentzian-like solutions of the scalar wave equation are deduced.