  The Hodge tau-function is a generating function for the linear Hodge
integrals. It is also a tau-function of the KP hierarchy. In this paper, we
first present the Virasoro constraints for the Hodge tau-function in the
explicit form of the Virasoro equations. The expression of our Virasoro
constraints is simply a linear combination of the Virasoro operators, where the
coefficients are restored from a power series for the Lambert W function. Then,
using this result, we deduce a simple version of the Virasoro constraints for
the linear Hodge partition function, where the coefficients are restored from
the Gamma function. Finally, we establish the equivalence relation between the
Virasoro constraints and polynomial recursion formula for the linear Hodge
integrals.
