Second order difference equations for certain families of `discrete' polynomials
The author describes two algorithms which can be used to obtain second order linear difference equations satisfied by certain families of discrete orthogonal polynomials. These algorithms can be implemented in any computer algebra system in order to obtain explicit expressions for the coefficients. The first algorithm deals with generalizations \(\{\widetilde{P}_n(x)\}\) of classical discrete orthogonal polynomials \(\{P_n(x)\}\) which can be expressed in the form \[ q(x;n)\widetilde{P}_n(x)=a(x;n)P_n(x)+b(x;n)P_n(x+1) \] and the second one with generalizations of the form \[ q(x;n)\widetilde{P}_n(x)=a(x;n)P_n(x)+b(x;n)P_{n-1}(x). \] Examples of these generalizations \(\{\widetilde{P}_n(x)\}\) are the so-called discrete Krall-type polynomials which are orthogonal with respect to the classical discrete weight function with one or two extra mass point(s) or Dirac delta measures in the endpoint(s) of the interval of orthogonality. In the last section of the paper two explicit examples are given for generalizations of the classical Meixner polynomials.
scientific article; zbMATH DE number 1326354