From Conformal Group to Symmetries of Hypergeometric Type Equations

Abstract

We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schr\"odinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.Comment: comment by PM: the editing done by SIGMA deteriorates the quality of the paper which is very sad, please visit http://www.fuw.edu.pl/~derezins/sym-hyp-submitted.pdf for the submitted and reviewed version which looks as the authors intended; part of PM's doctoral thesi