Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials

Abstract

Polynomials Qn(z), n = 0, 1, . . . , that are multi-orthogonal with respect to a Nikishin system of p ≥ 1 compactly supported measures over the star-like set of p + 1 rays S+ := {z ∈ C : z p+1 ≥ 0} are investigated. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order p + 1 of the form zQn(z) = Qn+1(z) + an Qn−p(z) with an > 0 for all n ≥ p, and that the nonzero roots of Qn are all simple and located in S+. Under the assumption of regularity (in the sense of Stahl and Totik) of the measures generating the Nikishin system, we describe the asymptotic zero distribution and weak behavior of the polynomials Qn in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms is proven.

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials

“…Some of the basic questions involve uniqueness of the MOP [7], convergence of the corresponding Hermite-Padé approximants [4], nth-root [8], ratio [3] (see also [13]), and strong [1,16] asymptotics of sequences of MOP. We have limited to a short list of significant contributions, see also reference lists in [14,15]. This paper is devoted to the study of the ratio asymptotic behavior of MOP associated with Nikishin systems of measures on star-like sets and of the limit behavior of the coefficients in the recurrence relation they satisfy.…”

“…In [15,Theorem 5.3,Corollary 5.4], under appropriate assumptions on the generating measures, the asymptotic zero distribution and nth-root asymptotics of the sequences (Q n ) ∞ n=0 and (Ψ n,k ) ∞ n=0 , k = 1, . .…”

“…We briefly recall some results from [14] that will be needed. As in [15,14], in this paper we will frequently use the notation [n :…”

“…The functions of the second kind satisfy the following recurrence relations. For every n ≥ p, 0 ≤ k ≤ p, we have [15,Proposition 3.2] zΨ n,k (z) = Ψ n+1,k (z) + a n Ψ n−p,k (z), and if n ≡ ℓ mod (p + 1), 0 ≤ ℓ ≤ p − 1, then ψ n,k (z) = ψ n+1,k (z) + a n ψ n−p,k (z), (2.4) while if n ≡ p mod (p + 1), then zψ n,k (z) = ψ n+1,k (z) + a n ψ n−p,k (z).…”

“…In [15,Theorem 3.5] it was proved that for n ≡ k mod p, 0 ≤ k ≤ p − 1, n ≥ p, the recurrence coefficient a n satisfies a n = K 2 n−p,k K 2 n,k .…”