Search for solutions to linear differential-difference equations in the degenerate case. (Russian) Zbl 1005.34078

Here, special solutions to the linear differential-difference equation \[ L(y)\equiv\sum^{n}_{i=0}\sum^{m}_{k=0}(a_{ik}x+b_{ik})y^{(i)}(x+h_{k})=F(x) \] are investigated, where \(x\) is a complex variable, the differences \(h_{k}\) are real numbers ( \(h_{0}<h_{1}<\dots <h_{m}\)) and the function \(F(x)\) can be presented by the Laplace integral. It is proved for the so-called degenerated case ( \(a_{n0}a_{nm}=0\)) that the above equation can have entire or \(\Gamma\)-similar solutions. A way of their construction is shown.


34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34K05 General theory of functional-differential equations
30D15 Special classes of entire functions of one complex variable and growth estimates