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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 $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of this recursion to the generalized Laguerre polynomial case.