We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det
M|^{2\alpha} e^{-N \Tr V(M)}dM$, where $\alpha>-1/2$ and $V$ is such that the
limiting mean eigenvalue density for $n,N\to\infty$ and $n/N\to 1$ vanishes
quadratically at the origin. In order to compute the double scaling limits of
the eigenvalue correlation kernel near the origin, we use the Deift/Zhou
steepest descent method applied to the Riemann-Hilbert problem for orthogonal
polynomials on the real line with respect to the weight
$|x|^{2\alpha}e^{-NV(x)}$. Here the main focus is on the construction of a
local parametrix near the origin with $\psi$-functions associated with a
special solution $q_\alpha$ of the Painlev\'e II equation $q''=sq+2q^3-\alpha$.
We show that $q_\alpha$ has no real poles for $\alpha > -1/2$, by proving the
solvability of the corresponding Riemann-Hilbert problem. We also show that the
asymptotics of the recurrence coefficients of the orthogonal polynomials can be
expressed in terms of $q_\alpha$ in the double scaling limit.