Laplace contour integrals and linear differential equations (Q2129476)

The author proves that to each linear differential equation \[ L[w]:=w^{(n)}+\sum\limits_{j=0}^{n-1} (a_j+b_jz) w^{(j)}=0, \quad a_0+b_0z \not\equiv 0 \] there exists some distinguished Laplace contour integral solution \[ \Lambda_L(z)=\frac{1}{2 \pi i} \int\limits_{\mathfrak{C}} \phi(t) e^{-zt} dt \] with kernel \(\phi\) that is uniquely determined by the operator \(L\) and itself determines the canonical contours \(\mathfrak{C}\). The main properties of these solutions are revealed: the order of growth, the deficiency of the value zero, asymptotic expansions in particular sectors, the distribution of zeros, the Phragmén-Lindelöf indicator, the Nevanlinna functions \(T(r,\Lambda_L)\) and \(N(r,1/\Lambda_L)\).