Creation and Annihilation Operators for Orthogonal Polynomials of Continuous and Discrete Variables

We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that correspond to the normalized polynomials and study their algebraic properties in the case of the Kravchuk/Hermite Meixner/Laguerre polynomials. 1. Introduction. In a previous paper [1] we have developed a method to construct raising and lowering operators for the Kravchuk polynomials of a discrete variable , using the properties of Wigner functions, and to calculate the continuous limit to the creation and annihilation operators for the solutions of the quantum harmonic oscillator. In this contribution we apply the same method to other orthogonal polynomi-als of discrete and continuous variable. We give general formulas for all orthogonal polynomials of hypergeometric type [2]: difference/differential equations, recurrence relations, raising and lowering operators. With the help of standard values we calculate these equations for the normalized functions of Kravchuk-Wigner and Meixner-Laguerre polynomials, and we construct the corresponding creation and annihilation operators. The motivation of this work is to implement the study of the Sturm-Liouville problem in continuous case with the discrete one, in particular the connexion between the eigenfunctions and the creation and annihilation operators [3] and [4]. This approach is becoming very powerful in the lattice formulation of field theories, where the physical properties of the model are analyzed in the lattice before the continuous limit is taken [5]