One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$, $$\int_z^\infty (\zeta-z)^{\alpha-1}f(\zeta) d\zeta = \lambda f(z) $$ where $\alpha$ is a constant with Re$(\alpha)>0$.
I found it by looking first at positive integer $\alpha$.
My question is: How does one solve for this eigenfunction problem in general and are there more eigenfunctions and eigenvalues?
I have tried Laplace and Fourier transforms. The integration limits and function growth rate block their respective applicability in the most straightforward manner --- that does preclude clever ways of application that circumvent these obstacles.