  We show that properties of hypergeometric class equations and functions
become transparent if we derive them from appropriate 2nd order differential
equations with constant coefficients. More precisely, properties of the
hypergeometric and Gegenbauer equation can be derived from generalized
symmetries of the Laplace equation in 4, resp. 3 dimension. Properties of the
confluent, resp. Hermite equation can be derived from generalized symmetries of
the heat equation in 2, resp. 1 dimension. Finally, the theory of the ${}_1F_1$
equation (equivalent to the Bessel equation) follows from the symmetries of the
Helmholtz equation in 2 dimensions. All these symmetries become very simple
when viewed on the level of the 6- or 5-dimensional ambient space. Crucial role
is played by the Lie algebra of generalized symmetries of these 2nd order
PDE's, its Cartan algebra, the set of roots and the Weyl group. Standard
hypergeometric class functions are special solutions of these PDE's
diagonalizing the Cartan algebra. Recurrence relations of these functions
correspond to the roots. Their discrete symmetries correspond to the elements
of the Weyl group.
