The Laguerre transform of some elementary functions (Q793937)

The first author [Aequationes Math. 22, 73-85 (1981; Zbl 0462.44002)] has introduced the so called generalized Laguerre transformation \[ {\mathcal I}^{(\alpha)}[f](\nu)=\int^{\infty}_{0}e^{- t}t^{\alpha}L_{\nu}^{(\alpha)}(t)f(t)dt \] where \(L_{\nu}^{(\alpha)}\) are the Laguerre functions of the first kind. In the present paper the following generalized Laguerre transforms are calculated \[ {\mathcal I}^{(\alpha)}[t^{\lambda -1}e^{-pt^ r}\left\{ \begin{matrix} \sin bt^ r\\ \cos bt^ r\end{matrix} \right\}](\nu) \] under suitable conditions on the constants. The result is used to find a particular solution of the following differential equation: \(e^ tD(e^{-t}tD)y(t)=e^{-2t} (t>0)\).