On the properties of an orthonormalized singular polynomials set

The paper introduces a concept of singular polynomials, and their orthonormalized set is constructed. The recurrence formula for three neighbouring orthonormalized singular polynomials is given and an analogue of the Rodrigues formula is found. It is established that the derivatives of polynomials from the built set have a property, close to the property of orthogonality. A differential equation and a family of relations for the derivatives of orthonormalized singular polynomials have been obtained. The results are applicable to the construction and investigation of the finite element method for mathematical physics problems with a strong singularity of solution.