Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

Abstract

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.

Classical symmetric orthogonal polynomials of a discrete variable

“…[14]) and should be surely fruitful. Darboux transformations for symmetric discrete orthogonal polynomials could also be examined like in Ref.…”

Recurrence relations for discrete hypergeometric functions

“…The main idea is to use the connection of the wave functions with the classical discrete polynomials in a similar way as it was done in our previous paper [16] for the N -th dimensional oscillators and hydrogenlike atoms. This approach allows us to recover the relations obtained in [5,19,20] and also to obtain several new relations for the discrete polynomials and therefore for the associated (wave) functions in a constructive way. This can be extended to other exactly solvable models which involve discrete hypergeometric functions or polynomials.…”

“…Methods for obtaining such recurrence relations have attracted the interest of several authors (see e.g. [19,20] and references therein).…”

Forward (&#916) and Backward (&#8711) Difference Operators Basic Sets of Polynomials in <img src="http://file.scirp.org/html/70175_c1.jpg" width="20"> and Their Effectiveness in Reinhardt and Hyperelliptic Domains

“…These operators are very important as they involve the discrete scheme used in numerical analysis. Furthermore, their composition operators form the most of second order difference equations of Mathematical Physics, the solutions of which are orthogonal polynomials [25] [26].…”