A purely algebraic derivation of associated Laguerre polynomials (Q6596140)

Classical orthogonal polynomials play a fundamental role in quantum mechanics, linear representation of Lie groups and in many other applications. As concerns the derivation of these important polynomials, there exist several methods including the Rodrigues formula, the use of a recurrence relation, solving a hypergeometric type equation satisfied by them, etc. Directly related to the classical orthogonal polynomials there are the corresponding associated special functions satisfying a differential equation which can be transformed into a Schrödinger type equation by using a change of variable. The equation satisfied by the associated Legendre functions corresponds to the radial part of the hydrogen-like atom Schrödinger equation. The algebraic methods based on the factorization method and the use of raising/lowering operators play an important role in the investigation of the properties of the associated special functions. Based on these techniques, the authors present an algebraic derivation of associated Laguerre polynomials. Their purely algebraic method admits as a starting point the knowledge of the energy eigenvectors of quantum mechanics solution of hydrogen-like atom Schrödinger equation.