On quadratic forms generated by entire functions (Q677729)

The following problem is investigated: let \(E_1, E_2, S_1\) and \(S_2\) be entire function; find functions \(D_1\) and \(D_2\) such that \(E_1D_2+E_2D_1=S_1S_2 \) holds on \(\mathbb{C}\). For any solution to this problem a corresponding entire function \(H\) of two variables is defined by \[ H(z,w)= (S_1(z)S_2(w)- E_1(z)D_2(z)- E_2(w)D_1(w))/(z-w). \] Let \((h_{kl})\) denote the Taylor coefficients of \(H\). The entire function \(H\) generates a quadratic form \(\kappa (x,y) = \sum_{k,l=0}^{\infty }x_ky_lh_{kl} \). In the first part of the paper the author solves the above problem under certain assumptions. In the second part the corresponding quadratic form \(\kappa \) is used to study solutions \(S\) of the operator equation \(AS-SB=Q,\) where \(E_1(A)=E_1(B)=0\).