On the Rodrigues’ Formula Approach to Operator Factorization

In this paper, we derive general formulae that reproduce well-known instances of recurrence relations for the classical orthogonal polynomials as special cases. These recurrence relations are derived, using only elementary mathematics, directly from the general Rodrigues’ formula for the classical orthogonal polynomials – a ‘first-principles’ derivation – and represent a unified presentation of various approaches to the exact solution of an important class of second-order linear ordinary differential equations. When re-expressed in ladder-operator form, the recurrence relations are seen to represent to a basic development of the work of Jafarizadeh and Fakhri [5] and allow a ‘Schrödinger operator factorization’ of the defining equation of the classical orthogonal polynomials, as well as an operational formula for the solution of this defining equation. The identity between the Rodrigues’ formula and the operational formula is determined and standard examples involving the application of the ladder-operator approach presented. The relationship with previous work is discussed