Symmetries and the compatibility condition for the new translational shape invariant potentials

TL;DR

This paper explores symmetries in a class of translational shape invariant potentials, showing a generalized compatibility condition aligns with shape invariance, and derives new relations for special functions.

Contribution

It demonstrates the equivalence of a generalized compatibility condition to shape invariance and introduces new relations for special functions related to these potentials.

Findings

Generalized compatibility condition is equivalent to shape invariance.

New relations for Laguerre, Jacobi, and hypergeometric functions.

Analysis of recent polynomial and rational extensions of shape invariant potentials.

Abstract

In this letter we study a class of symmetries of the new translational extended shape invariant potentials. It is proved that a generalization of a compatibility condition introduced in a previous article is equivalent to the usual shape invariance condition. We focus on the recent examples of Odake and Sasaki (infinitely many polynomial, continuous $l$ and multi-index rational extensions). As a byproduct, we obtain new relations, to the best of our knowledge, for Laguerre, Jacobi polynomials and (confluent) hypergeometric functions.