ON RECURRENCE RELATIONS FOR SOBOLEV ORTHOGONAL POLYNOMIALS

This paper discusses recurrence relations for sequences of polynomials which are orthogonal with respect to the Sobolev inner product defined on the set of polynomials P by GRAPHICS for some integer N greater than or equal to 1, where each mu(k), 0 less than or equal to k less than or equal to N, is a positive Borel measure. It is proven that there exists a real-valued polynomial h:R --> R satisfying (*) (hp,q)w = (p,hq)w (p,q is an element of P) if and only if each of the measures mu(k),1 less than or equal to Ic I N, is purely atomic with a finite number of mass points. In addition it is proven that R(j), the set of real roots of d(3)h/dx(3), (1 less than or equal to j less than or equal to N), is nonempty and that supp (mu(k)) subset of boolean AND(i=1)(k)R(i). It is also shown that if h satisfies the condition (*), then the polynomials orthogonal with respect to the inner product (.,.)w will satisfy a recurrence relation of order 2m fl, where m = deg(h). Furthermore, an algorithm is given to construct a polynomial H of minimal positive degree for which the above properties hold. Several examples will be discussed to illustrate the theory. Lastly it is shown, under certain circumstances, when these orthogonal polynomials will satisfy second-order linear differential equations.