Generalized associated polynomials and functions of second kind

Abstract

Let { p ν } ν ∈ N 0 , p ν ∈ Π ν ⧹ Π ν - 1 , be a sequence of polynomials, generated by a three-term recurrence relation. Shifting the recurrence coefficients of the elements of { p ν } ν ∈ N 0 we get a sequence of so-called associated polynomials, which play an important role in the theory of orthogonal polynomials. We generalize this concept of associating for arbitrary polynomials v n ∈ Π n . Especially, if v n is expanded in terms of p ν , ν = 0 , … , n , their associated polynomials are Clenshaw polynomials which are used in numerical mathematics. As a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of Clenshaw polynomials. Analogously as for orthogonal polynomials we define functions of second kind for v n . We prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials.