From Kontsevich-Witten to linear Hodge integrals via Virasoro operators

We give a proof of Alexandrov's conjecture on a formula connecting the Kontsevich-Witten and Hodge tau-functions using only the Virasoro operators. This formula has been confirmed up to an unknown constant factor. In this paper, we show that this factor is indeed equal to one by investigating series expansions for the Lambert W function on different points.