  The multi-indexed Jacobi polynomials are the main part of the eigenfunctions
of exactly solvable quantum mechanical systems obtained by certain deformations
of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the
parameter(s) of the P\"oschl-Teller potential, we obtain several families of
explicit and global solutions of certain second order Fuchsian differential
equations with an apparent singularity of characteristic exponent -2 and -1.
They form orthogonal polynomials over $x\in(-1,1)$ with weight functions of the
form $(1-x)^\alpha(1+x)^\beta/\{(ax+b)^4q(x)^2\}$, in which $q(x)$ is a
polynomial in $x$.
