Inverse Laplace transform of $\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p$

Question

I am hoping to identify the function $f(t)$ that has the following Laplace transform,

$$ \tilde f(s)=\int_0^\infty f(t)e^{-st}dt=\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p $$

where $\alpha,s_0,p$ are positive parameters. Any suggestions about how to approach this problem?

Answer 1

Using Leucippus's hint, we have $$\mathcal L^{-1} \!\left[ \frac 1 s \left( 1 + \frac 1 s \right)^p \right] = L_p(-t), \\ \mathcal L^{-1} \!\left[ \left( 1 + \frac 1 s \right)^p \right] = \frac d {d t} L_p(-t) = L_{p - 1}^{(1)}(-t) + \delta(t), \\ \mathcal L^{-1} \!\left[ \left( \frac {1 + \alpha s} {1 + \alpha (s - s_0)} \right)^p \right] = \mathcal L^{-1} \!\left[ \left( 1 + \frac 1 {s/s_0 + 1/(\alpha s_0) - 1} \right)^p \right] = \\ s_0 e^{(s_0 - 1/\alpha)t} L_{p - 1}^{(1)}(-s_0 t) + \delta(t),$$ where $L_{p - 1}^{(1)}$ is the generalized Laguerre function.