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The quantum spectral curve equation associated to KP $\tau$-functions ofhypergeometric type serving as generating functions for rationally weightedHurwitz numbers is solved by generalized hypergeometric series. The basiselements spanning the corresponding Sato Grassmannian element are shown to beMeijer $G$-functions, or their asymptotic series. Using their Mellin integralrepresentation the $\tau$-function, evaluated at the trace invariants of anexternally coupled matrix, is expressed as a matrix integral.

Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals