Treffer: Functions without residue and a bilinear differential equation

Let \(K\) be a field of zero characteristic. A rational function \(w\in K(X)\) is called special, if all residues of \(w\) and \(1/w\) vanish. It is called superspecial, if all residues of \(w^ n\) vanish for \(n\in\mathbb{Z}\). A function is called simple, if it has the form \(cf^ 2/g^ 2\), where \(c\) is a constant and \(f\), \(g\) are products of distinct linear functions. The author obtains several interesting results dealing with these classes of functions and in particular shows that if \((f,g)=1\), then \((f/g)^ 2\) is special and simple if and only if \(f''g-2f'g'+fg''=0\). Moreover it is shown that every superspecial function is of the form \(c(x-\alpha)^ n\) with non-zero \(c\), \(n\in\mathbb{Z}\), \(n\neq\pm1\).