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Eigenfunctions of Laguerre-type operators and generalized evolution problems. (English) Zbl 1097.34551
Setting $D=d/dx$, eigenfunctions of the following Laguerre-type operators $$D^{h+1}x^jD^{j-h}\text{ and } D^{p_1}x^{q_1}\cdots D^{p_r}x^{q_r}D^{p_{r+1}},$$ are explicitly given, where $h, j\ge 0, p_1, \cdots, p_{r+1}, q_1, \cdots, q_r>0$ are integers such that $j>h$, $p_1+\cdots+p_{r+1}=q_1+\cdots +q_r+1$. As an application, an operational solution of the evolution problem $$\hat{\Omega}_xS(x,t)=\frac{\partial}{\partial t} S(s, t)\quad \text{ in the half-plane } x>0,\quad K \lim_{x\to 0+}x^{-H}S(x,t)=s(t),$$ is obtained, where $\widehat\Omega_x$ is a differential operator in $x$, $H$, $K>0$ are constants and $s(t)$ is an analytic function.

##### MSC:
 34L40 Particular ordinary differential operators 47E05 Ordinary differential operators
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##### References:
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