Fourt-order differential equation satisfied by the associated of any order of all classical orthogonal polynomials. A study of their distribution of zeros

AbstractThe first associated (numerator polynomials) of all classical orthogonal polynomials satisfy one fourth-order differential equation valid for the four classical families, but for the associated of arbitrary order the differential equations are only known separately. In this work we introduce a program built in Mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class. Then we use this differential equation in order to study the distribution of zeros of these polynomials via their Newton sum rules (i.e., the sums of the kth power of zeros) which are closely related with the moments of such a distribution