A criterion for the solvability of the multiple interpolation problem by simple partial fractions

Using reduction to polynomial interpolation, we study the multiple interpolation problem by simple partial fractions. Algebraic conditions are obtained for the solvability and the unique solvability of the problem. We introduce the notion of generalized multiple interpolation by simple partial fractions of order $\le n$. The incomplete interpolation problems (i.e., the interpolation problems with the total multiplicity of nodes strictly less than $n$) are considered; the unimprovable value of the total multiplicity of nodes is found for which the incomplete problem is surely solvable. We obtain an order $n$ differential equation whose solution set coincides with the set of all simple partial fractions of order $\le n$.