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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 Pöschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of the Pöschl-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 $(1-x)^α(1+x)^β/\{(ax+b)^4q(x)^2\}$, in which $q(x)$ is a polynomial in $x$.