  Using Atiyah-Bott localization on the space of stable maps to the stack
quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all
Hodge integrals with descendent insertions at one marked point on the
hyperelliptic locus $\overline{\mathcal{H}}_{g, 2g + 2} \subseteq
\overline{\mathcal{M}}_{g, 2g + 2}$. The initial conditions required for our
recursions are gravitational descendents at one marked point, which are known
to be $\frac{1}{2}$. We discover a new structure concerning these intersection
numbers: for a fixed monomial of $\lambda$-classes, the resulting family of
hyperelliptic Hodge integrals is polynomial in $g$. We formulate a conjecture
concerning the log-concavity of the coefficients of these polynomials. Lastly,
we turn our recursions into a non-linear system of partial differential
equations for the generating functions of hyperelliptic Hodge integrals.
