Bergman operators are linear
integral operators that map complex analytic functions into solutions of linear partial
differential equations with analytic coefficients. In this way methods and
results of complex analysis can be used for characterizing general properties
of classes of those solutions. For example, this approach yields theorems
about the location and type of singularities, the growth, and the coefficient
problem for series developments of solutions. A partial differential equation
being given, there exist various types of Bergman operators, and for that
purpose it is essential to select an operator whose generating function is as
simple as possible. The present paper considers differential equations in two
independent variables, introduces a class of Bergman operators satisfying
that requirement, and determines the corresponding class of differential
equations in an explicit fashion. In fact, necessary and sufficient conditions
are obtained in order that the solutions of a partial differential equation
can be obtained by means of a Bergman operator of that class. It is also
shown that the set of these equations includes several equations of practical
importance.
