On the modifications of semi-classical orthogonal polynomials on nonuniform lattices

Semi classical orthogonal polynomials on nonuniform lattices with respect to a linear functional LL are defined as polynomials (Pn)(Pn) where the degree of PnPn is exactly n  , the PnPn satisfy the orthogonality relation 〈L,PnPm〉=0,n≠m,〈L,PnPn〉≠0,n≥0 and LL satisfies the Pearson equation Dx(ϕL)=Sx(ψL),Dx(ϕL)=Sx(ψL), where ϕ is a non zero polynomial and ψ a polynomial of degree at least 1. In this work, we prove that the multiplication of semi classical linear functional by a first degree polynomial, the addition of a Dirac measure to the semi-classical regular linear functional on nonuniform lattice give semi classical linear functional but not necessary of the same class. We apply these modifications to some classical orthogonal polynomials.