Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials

TL;DR

This paper extends the study of pseudo-Wronskian determinants to Laguerre and Jacobi polynomials, revealing richer equivalence relations linked to shape invariance and discrete symmetries in quantum potentials.

Contribution

It derives new equivalence relations for Laguerre and Jacobi pseudo-Wronskians, incorporating four seed function families and connecting to shape invariance in quantum systems.

Findings

Richer equivalence relations for Laguerre and Jacobi pseudo-Wronskians.

Dependence on two Maya diagrams for these polynomials.

Interpretation as shape invariance and symmetries in quantum potentials.

Abstract

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Poschl-Teller potential.