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

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.

► We study a class of symmetries of the new translational shape invariant potentials.
► We prove that the compatibility and shape invariance conditions are equivalent.
► We apply the results to the recent examples of Odake and Sasaki.
► We obtain new relations for special functions.