Another set of infinitely many exceptional (X_{ell}) Laguerre polynomials

We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one dimensional quantum mechanics and the corresponding X_{\ell} Laguerre and Jacobi polynomials (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417). The new X_{\ell} Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X_{\ell} Jacobi polynomials and the potentials, whereas the known X_{\ell} Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known X_{\ell} Jacobi polynomials and the potentials.

💡 Research Summary

This paper expands the family of exactly solvable one‑dimensional quantum‑mechanical models by introducing a third infinite set of shape‑invariant potentials together with the corresponding exceptional (Xℓ) Laguerre polynomials. The motivation stems from the work of Odake and Sasaki (Phys. Lett. B 679 (2009) 414‑417), who constructed two infinite families of shape‑invariant potentials and associated Xℓ Laguerre and Xℓ Jacobi polynomials. Those families were obtained by taking limits of the ordinary Laguerre and Jacobi systems, respectively, and by using a mirror‑image transformation of the Jacobi parameters. The present contribution shows that a further infinite family can be generated directly from the known Xℓ Jacobi polynomials by a simple limiting procedure, without invoking the mirror image.

The construction proceeds as follows. Starting from the Xℓ Jacobi polynomials (P^{(\alpha,\beta)}_{\ell,n}(x)) (with parameters (\alpha,\beta> -1) and variable (x\in