## 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.

### MSC:

 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

### Keywords:

solutions; linear differential-difference equation