On associated and co-recursive d-orthogonal polynomials

Abstract

Abstract The associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given.