  We study the Jacobi unitary ensemble perturbed by an algebraic singularity at
$t>1$. For fixed $t$, this is the modified Jacobi ensemble studied by Kuijlaars
{\it{et al.}} The main focus here, however, is the case when the algebraic
singularity approaches the hard edge, namely $t\to 1^+$.
  In the double scaling limit case when $t- 1$ is of the order of magnitude of
$1/n^2$, $n$ being the size of the matrix, the eigenvalue correlation kernel is
shown to have a new limiting kernel at the hard edge $1$, described by the
$\psi$-functions for a certain second-order nonlinear equation. The equation is
related to the Painlev\'e III equation by a M\"obius transformation. It also
furnishes a generalization of the Painlev\'e V equation, and can be reduced to
a particular Painlev\'e V equation via the B\"acklund transformations in
special cases. The transitions of the limiting kernel to Bessel kernels are
also investigated, with $n^2(t-1)$ being large or small.
  In the present paper, the approach is based on the Deift-Zhou nonlinear
steepest descent analysis for Riemann-Hilbert problems.
