Meijer's $G$-function and Euler's differential equation revisited

We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer's $G$-functions. In the course of deriving the solution of the generalized Euler eigenvalue equation we review some of the basics of generalized hypergeometric functions and Meijer's $G$-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles.