  This is the first of a series of articles providing a foundation for the
theory of Drinfeld modular forms of arbitrary rank r. In the present part, we
develop the analytic theory. Most of the work goes into defining and studying
the u-expansion of a weak Drinfeld modular form, whose coefficients are weak
Drinfeld modular forms of rank r-1. Based on that we give a precise definition
of when a weak Drinfeld modular form is holomorphic at infinity and thus a
Drinfeld modular form in the proper sense.
