The Harer-Zagier recursion for an irregular spectral curve

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms W1(g) of the genus expansion of the one-loop mean are polynomials. We generalize this recursion to the case of generalized Laguerre ensembles corresponding to unicellular bicolored maps with different weights of vertices of different colors.