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

Abstract

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 exponents −2 and −1. They form orthogonal polynomials over x ∈ ( − 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.