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

TL;DR

This paper explores the properties of multi-indexed Jacobi polynomials arising from deformations of the P"oschl-Teller potential, revealing explicit solutions to Fuchsian differential equations with apparent singularities and their orthogonality properties.

Contribution

It introduces new families of orthogonal polynomials derived from quantum mechanical systems with deformed potentials, highlighting their explicit solutions and singularity structures.

Findings

Explicit solutions to second order Fuchsian equations with apparent singularities.

Orthogonality of the new polynomial families over (-1,1).

Weight functions involving polynomial and rational factors.

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 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 $x$.