Eigenfunctions of Laguerre-type operators and generalized evolution problems (Q815493)

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 \(h, j\geq 0, p_1, \cdots, p_{r+1}, q_1, \cdots, q_r>0\) 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.