  The quantum spectral curve equation associated to KP $\tau$-functions of
hypergeometric type serving as generating functions for rationally weighted
Hurwitz numbers is solved by generalized hypergeometric series. The basis
elements spanning the corresponding Sato Grassmannian element are shown to be
Meijer $G$-functions, or their asymptotic series. Using their Mellin integral
representation the $\tau$-function, evaluated at the trace invariants of an
externally coupled matrix, is expressed as a matrix integral.
