Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case

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 $(1-x{)}^{\alpha }(1+x{)}^{\beta }\mathrm{/}\{(ax+b{)}^{4}q(x{)}^2\}$, in which $q(x)$ is a polynomial in $x$.