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.