Darboux-crum transformations of orthogonal polynomials and associated boundary conditions

Abstract

Linear second order ordinary di erential boundary value problems feature prominently in many scienti c eld, such as physics and engineering. Solving these problems is often riddled with complications though a myriad of techniques have been devised to alleviate these di culties. One such method is by transforming a problem into a more readily solvable form or a problem which behaves in a manner which is well understood. The Darboux-Crum transformation is a particularly interesting transformation characterised by some surprising properties, and an increase in the number of works produced in the last few years related to this transformation has prompted this investigation. The classical orthogonal polynomials, namely those of Jacobi, Legendre, Hermite and Laguerre, have been nominated as test candidates and this work will investigate how these orthogonal families are a ected when transformed via Darboux-Crum transformations.