2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/137817In this note, we symmetrized the cut-join equation from the proof of Marino-Vafa formula by applying a transcendental changing of variable. One can derive more recursion formulas of Hodge integrals out of this polynomial equations. We also give some applications.Mathematical PhysicsCombinatoricstext