Holomorphic automorphic differential operators on Siegel upper half space are important objects which have a lot of applications to number theory, but they are interesting as itself as a new theory of special functions, containing the classical theory of Gegenbauer functions.
We construct explicitly the universal automorphic differential operator which is a source of all such operators. This is a ultimate generalization of generating function of Gegenbauer polynomials to many variables. We also explain the associated holonomic (integrable) system.
| 
