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Transactions of the American Mathematical Society Asymptotic zero distribution for a class of multiple orthogonal polynomials
Asymptotic zero distribution for a class of multiple orthogonal polynomials
Coussement, E., Coussement, J., Van Assche, W.Bu kitabı nə dərəcədə bəyəndiniz?
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Transactions of the American Mathematical Society
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10.1090/s0002994708045352
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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 10, October 2008, Pages 5571–5588 S 00029947(08)045352 Article electronically published on May 20, 2008 ASYMPTOTIC ZERO DISTRIBUTION FOR A CLASS OF MULTIPLE ORTHOGONAL POLYNOMIALS E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE Abstract. We establish the asymptotic zero distribution for polynomials generated by a fourterm recurrence relation with varying recurrence coeﬃcients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular JacobiPiñeiro, Laguerre I and the example associated with modiﬁed Bessel functions. We also discuss an application to Toeplitz matrices. 1. Introduction Let µ be a positive measure on the real line for which the support is not ﬁnite and all the moments exist. The corresponding monic orthogonal polynomial Pn of degree n is then deﬁned by (1.1) xm Pn (x) dµ(x) = 0, k = 0, . . . , n − 1, with P0 ≡ 1 and P−1 ≡ 0. A wellknown fact is that such polynomials satisfy a threeterm recurrence relation of the form (1.2) zPn (z) = Pn+1 (z) + bn Pn (z) + a2n Pn−1 (z), an > 0, bn ∈ R, with initial conditions P0 ≡ 1 and P−1 ≡ 0. An object of frequent study is the asymptotic zero distribution of the zeros for a sequence of orthogonal polynomials. The zeros of the polynomials Pn , generated by (1.2), are real and simple [13]. With each polynomial Pn we can associate the normalized zero counting measure 1 δx , (1.3) ν(Pn ) := n Pn (x)=0 where δx is the Dirac point mass at x. If limn→∞ ν(Pn ) = ν, by which we mean that f dν(Pn ) = f dν lim n→∞ Received by the editors June 19, 2006 and, in revised form, January 31, 2007. 2000 Mathematics Subject Classiﬁcation. Primary 33C45, 42C05; Secondary 15A18. Key words and phrases. Multiple orthogonal polynomials, asymptotics. This work was supported by INTAS project 03516637, by FWO projects G.0455.04 and G.0184.02 and by OT/04/21; of Katholieke Universiteit Leuven. The second author is a postdoctoral researcher at the Katholieke Universiteit Leuven (Belgium). c 2008 American Mathematical Society Reverts to public domain 28 years from publication 5571 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5572 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE for every bounded and continuous function f on R (weak convergence), then we call the probability measure ν the asymptotic zero distribution of the sequence {Pn }n≥0 . One of the famous results in this context is the following. Theorem 1.1 (see, e.g., [24, 28]). Suppose that the recurrence coeﬃcients an > 0 and bn ∈ R have the limits a > 0 and b ∈ R, respectively. The polynomials Pn , generated by (1.2), then have the asymptotic zero distribution ω[γ,δ] with density ⎧ 1 ⎨ , x ∈ [γ, δ], dω[γ,δ] (x) = (1.4) π (δ − x)(x − γ) ⎩ dx 0, elsewhere, where γ = b − 2a and δ = b + 2a. Remark 1.2. The measure ω[γ,δ] is known as the arcsine measure on [γ, δ]. It also minimizes the logarithmic energy of the interval [γ, δ] [26]. Recently, the result in Theorem 1.1 was extended to the case of varying recurrence coeﬃcients. Here the notation limn/N →t Yn,N = Y denotes the property that in the doubly indexed sequence Yn,N we have limj→∞ Ynj ,Nj = Y whenever nj and Nj are two sequences of natural numbers such that Nj → ∞ and nj /Nj → t as j → ∞. Theorem 1.3 (Kuijlaars, Van Assche [21]). Let for each N ∈ N, two sequences ∞ {an,N }∞ n=1 , an,N > 0, and {bn,N }n=0 , bn,N ∈ R, of recurrence coeﬃcients be given. Furthermore, suppose there exist two continuous functions a : (0, +∞) → [0, +∞), b : (0, +∞) → R, such that (1.5) lim an,N = a(t), n/N →t lim bn,N = b(t), n/N →t t > 0, and deﬁne γ(t) := b(t) − 2a(t), δ := b(t) + 2a(t), t > 0. For the (orthogonal) polynomials generated by the recurrence (1.6) zPn,N (z) = Pn+1,N (z) + bn,N Pn,N (z) + a2n,N Pn−1,N (z), with initial conditions P0,N ≡ 1 and P−1,N ≡ 0, we then have 1 t ω[γ(s),δ(s)] ds, t > 0. (1.7) lim ν(Pn,N ) = t 0 n/N →t Here ω[γ,δ] is deﬁned by (1.4) if γ < δ and by δγ if γ = δ. Remark 1.4. More recently, Theorem 1.3 was generalized to measurable functions a and b [20]. In this paper we present a (conditional) theorem giving the asymptotic zero distribution for polynomials satisfying a fourterm recurrence relation of the form (1.8) zPn,N (z) = Pn+1,N (z) + bn,N Pn,N (z) + cn,N Pn−1,N (z) + dn,N Pn−2,N (z), where the varying recurrence coeﬃcients have some particular limiting behavior. So, in a sense it extends Theorem 1.3. Such a fourterm recurrence relation appears in the theory of multiple orthogonal polynomials of Type II. These are a generalization of orthogonal polynomials which arises naturally in HermitePadé approximation of a system of (Markov) functions [9, 10, 22]. In particular, they satisfy orthogonality conditions with respect to several positive measures [2, 25, 29]. Some of their applications are situated in diophantine number theory, rational approximation, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS 5573 spectral and scattering problems for higherorder diﬀerence equations and some associated dynamical systems; see, e.g., [5, 11, 18, 27]. Recently they also appeared in random matrix theory for matrix ensembles with external source [3, 7, 8] and Wishard ensembles [6]. The particular limiting behavior which we are considering appears in the examples JacobiPiñeiro, Laguerre I [29] and the example associated with modiﬁed Bessel functions [30]. In Subsection 2.1 we state our main theorem. Next, in Subsection 2.2 and Subsection 2.3 we apply this result to the examples of multiple orthogonal polynomials mentioned above and some particular kind of Toeplitz matrices. In Section 3 we discuss a theorem on ratio asymptotics for monic polynomials satisfying the recurrence (1.8). This will be used to prove our main theorem in Section 4. 2. Statement of results 2.1. Main theorem. We will study doubly indexed sequences of polynomials {Pn,N }, generated by a fourterm recurrence of the form (2.1) zPn,N (z) = Pn+1,N (z) + bn,N Pn,N (z) + cn,N Pn−1,N (z) + dn,N Pn−2,N (z), with the initial conditions P0,N ≡ 1, P−1,N ≡ 0 and P−2,N ≡ 0 and real recurrence coeﬃcients. In particular, our main theorem gives an explicit expression for the asymptotic zero distributions lim ν(Pn,N ), n/N →t t > 0, with some conditions on the zeros of the Pn,N and some particular limiting behavior for the recurrence coeﬃcients. As mentioned in the introduction the limit is taken over any sequence {ν(Pnj ,Nj )}j≥1 for which nj → ∞, Nj → ∞ and nj /Nj → x as j → ∞. We will use this notation throughout the rest of this paper. ∞ Theorem 2.1. Let for each N ∈ N three sequences {bn,N }∞ n=0 , {cn,N }n=1 and ∞ {dn,N }n=2 of real recurrence coeﬃcients be given and assume that there exists a continuous function α : [0, +∞) → [0, +∞) such that, for t > 0, (2.2) lim bn,N = 3β(t), n/N →t lim cn,N = 3β(t)2 , n/N →t lim dn,N = β(t)3 , n/N →t with β(t) = 4α(t) 27 . Let Pn,N be the monic polynomials generated by the recurrence (2.1) and suppose these polynomials Pn,N have real simple zeros xn,N < . . . < xn,N n 1 n+1,N n,N n+1,N satisfying the interlacing property xj < xj < xj+1 , for all n, N ∈ N, j = 1, . . . , n. Then 1 t (2.3) lim ν(Pn,N ) = υ[0,α(s)] ds, t > 0, t 0 n/N →t where υ[0,α] is deﬁned ⎧ ⎪ ⎨ dυ[0,1] (x) = (2.4) ⎪ dx ⎩ dυ dυ [0,α] [0,1] x by δ0 if α = 0 and dx (x) = α1 dx ( α ), with √ √ √ 3 (1 + 1 − x)1/3 + (1 − 1 − x)1/3 √ , x ∈ (0, 1), 4π x2/3 1 − x 0, elsewhere, if α > 0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5574 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 Figure 1. The density of the measure υ[0,1] . Remark 2.2. Denote by νt the right hand side of (2.3). The action of this measure on arbitrary Borel sets is given by 1 t υ[0,α(s)] (E) ds, E ∈ B(R). νt (E) = t 0 Now suppose that for each x ≥ 0 the set {s ≥ 0  x ≤ α(s)} is an interval, which we denote by [t− (x), t+ (x)]. The density of the measure νt is then dνt 1 min(t,t+ (x)) dυ[0,α(s)] (2.5) (x) = (x) ds. dx t min(t,t− (x)) dx This will be the case in each of the examples we present in this paper. Remark 2.3. Comparing Theorem 2.1 with Theorem 1.3 we see that the measure υ[0,1] plays the role of the arcsine measure in the case of orthogonal polynomials (satisfying a threeterm recurrence relation). The density (2.4) again has the behavior c1 (1 − x)−1/2 as x ↑ 1, but has a diﬀerent behavior c2 x−2/3 as x ↓ 0. See Figure 1. Remark 2.4. The measure υ[0,1] coincides (after a cubic transformation) with the asymptotic zero distribution of Faber polynomials associated with the 3cusped hypocycloid [19]. 2.2. Application to multiple orthogonal polynomials. There are two types of multiple orthogonal polynomials, but we will only consider type II. Let µ1 , . . . , µr , r ∈ N, be a set of positive measures on the real line for which the support is not ﬁnite and all the moments exist. Furthermore, let n = (n1 , n2 , . . . , nr ) be a vector of r nonnegative integers, which is a multiindex with length n := n1 + n2 + · · · + nr . A multiple orthogonal polynomial Pn of type II with respect to the multiindex n, is a (nontrivial) polynomial of degree ≤ n which satisﬁes the orthogonality conditions 0 ≤ m ≤ nj − 1, j = 1, . . . , r. (2.6) xm Pn (x) dµj (x) = 0, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS 5575 A basic requirement in the study of multiple orthogonal polynomials is that the system (2.6) has a unique solution (up to a scalar multiplicative constant) of degree n. We call n a normal index for µ1 , . . . , µr if any solution of (2.6) has exactly degree n (which implies uniqueness). If all the multiindices are normal, then the system of measures is called perfect. Some famous classes of perfect systems are the Angelesco systems, Nikishin systems (for r = 2) and AT systems; see, e.g., [25, 29]. Multiple orthogonal polynomials of type II satisfy a recurrence relation of order r+1. In particular, if we set r = 2 and consider proper multiindices νn = (m+s, m), n ∈ N ∪ {0}, where n = 2m + s, s ∈ {0, 1}, then the polynomials Pn := Pνn satisfy a fourterm recurrence relation of the form (2.7) zPn (z) = Pn+1 (z) + bn Pn (z) + cn Pn−1 (z) + dn Pn−2 (z), with the initial conditions P0 ≡ 1, P−1 ≡ 0 and P−2 ≡ 0. For three examples known in the literature the recurrence coeﬃcients in (2.7) have the particular limiting behavior (2.2), possibly after some rescaling. In each of these examples the measures form an AT system on an interval ∆ ⊆ R. It is then known that the zeros of the polynomials Pn are simple, lie in ∆ [25, 29] and satisfy the interlacing property [4]. So, it is possible to apply Theorem 2.1. 2.2.1. JacobiPiñeiro. The JacobiPiñeiro polynomials are the multiple orthogonal polynomials for the system of orthogonality measures dµj (x) = xαj (1 − x)β dx, j = 1, 2, / Z. In [29] it was shown that on the interval [0, 1] with α1 , α2 , β > −1 and α2 − α1 ∈ the monic JacobiPiñeiro polynomials with respect to proper multiindices, which we denote by Pnα1 ,α2 ;β , satisfy a recurrence relation of the form (2.7) for which 2 3 4 4 4 (2.8) lim bn = 3 , lim dn = . , lim cn = 3 n→∞ n→∞ n→∞ 27 27 27 By Theorem 2.1 with α(t) = 1, t > 0, we then easily obtain the following result. Theorem 2.5. The JacobiPiñeiro polynomials Pnα1 ,α2 ;β have the asymptotic zero distribution υ[0,1] , deﬁned as in (2.4). 2.2.2. Multiple Laguerre I. The multiple Laguerre polynomials of the ﬁrst kind are orthogonal with respect to the system of measures dµj (x) = xαj e−x dx, j = 1, 2, on [0, +∞) with α1 , α2 > −1 and α2 −α1 ∈ / Z. Denote the monic multiple Laguerre I 1 ,α2 polynomials with respect to proper multiindices by Lα . These satisfy a fourn term recurrence relation of the form (2.7) where, for t > 0, 2 3 t t t bn cn dn (2.9) lim =3 =3 , lim = ; , lim 2 2 2 n/N →t N n/N →t N 2 n/N →t N 3 see [29]. The following theorem is then a corollary of Theorem 2.1. Theorem 2.6. For the multiple Laguerre polynomials of the ﬁrst kind the limit 1 (2.10) νtL := lim δx/N , t > 0, n/N →t n α ,α 1 2 Ln (x)=0 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5576 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE 4 Laguerre I Mcdonald 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 Figure 2. The densities of the measures νtL and νtM , see (2.11) 8 2 and t = 3√ , respectively. and (2.14), with t = 27 3 exists and has the density (2.11) dνtL (x) = dx 8 27t g 8x 27t x ∈ (0, 27t 8 ), , 0, elsewhere, where, for y ∈ (0, 1), √ √ √ √ √ 3 3 (1 + 3 1 − y)(1 − 1 − y)1/3 − (1 − 3 1 − y)(1 + 1 − y)1/3 g(y) = . 16π y 2/3 1 ,α2 1 ,α2 Proof. If we deﬁne L̃α (z) := Lα (N z)/N n , then the polynomials satisfy a n n recurrence relation of the form (2.1) and the asymptotic property (2.2) with α(t) = 27t 8 , t > 0. So, applying Theorem 2.1 we get dνtL 1 (x) = dx t t 8 dυ[0,1] 27s dx 8x 27s 1 8 ds = t 27 8x 27 1 1 dυ[0,1] (u) du. u dx 8x 27t √ 8x Set x ∈ (0, 27t 1 − 27t . Applying the substitution y ↔ 1 − u we then 8 ) and z = obtain √ dνtL 14 3 z (1 + y)−4/3 (1 − y)−5/3 + (1 + y)−5/3 (1 − y)−4/3 dy (x) = dx t 27π 0 √ 14 3 z = (1 + y)−5/3 (1 − y)−4/3 dy t 27π −z z √ 1 3 1 + 3y = . t 18π (1 + y)2/3 (1 − y)1/3 −z This completes the proof. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS 5577 2.2.3. Multiple orthogonal polynomials associated with modiﬁed Bessel functions. In [30] one considered multiple orthogonal polynomials with respect to the orthogonality measures dµ1 (x) = xκ ργ (x) dx, dµ2 (x) = xκ ργ+1 (x) dx on (0, +∞) with κ > −1, γ ≥ 0 and √ ργ (x) = 2xγ/2 Kγ (2 x), x > 0, where Kγ is the modiﬁed Bessel of the second kind, also known as the Macdonald function [1, p. 374]. For the type II multiple polynomials with respect to proper multiindices, Pnγ;κ , the recurrence coeﬃcients in (2.7) are known [30, Theorem 4]. In particular, for t > 0, (2.12) lim n/N →t bn = 3t2 , N2 lim n/N →t cn = 3t4 , N4 lim n/N →t dn = t6 . N6 Theorem 2.1 then implies the following asymptotic result for the zeros of these polynomials. Theorem 2.7. For the multiple orthogonal polynomials associated with modiﬁed Bessel functions the limit 1 (2.13) νtM := lim δx/N 2 n/N →t n γ;κ Pn (x)=0 exists and has the density (2.14) where ⎧ ⎨ 42 h dνtM 27t (x) = ⎩ 0, dx 4x 27t2 , 2 x ∈ (0, 27t 4 ), elsewhere, √ √ √ 3 3 (1 + 1 − y)1/3 − (1 − 1 − y)1/3 h(y) = , 4π y 2/3 y ∈ (0, 1). Proof. Deﬁne the polynomials P̃nγ;κ (z) := Pnγ;κ (N 2 z)/N 2n . By (2.12) these satisfy a recurrence relation of the form (2.1) having the asymptotic property (2.2) with 2 27t2 4x α(t) = 27t 4 , t > 0. Set x ∈ (0, 4 ) and z = 27t2 . Applying Theorem 2.1 then gives t 1 4x 4 dυ[0,1] dνtM (x) = 2 dx t √ 27s dx 27s2 2 x 3 1 2 ds = 2 √ t 27 z 1 1 dυ[0,1] √ (u) du. u dx z 3 √ Similarly as in Theorem 2.6 we apply the substitution y ↔ 1 − u and get √ √1−z dνtM 1 3 1 √ (x) = 2 (1 + y)−5/6 (1 − y)−7/6 dy dx t 27π z −√1−z √ 1/6 1−z 1+y 1 1 1 √ √ . = √ t2 3 3π z 1 − y − 1−z From this we easily obtain (2.14). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5578 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE 2.3. Application to Toeplitz matrices. Set Toeplitz matrices ⎛ 3β 1 0 ... ... ⎜ .. ⎜ 3β 2 3β . 1 ⎜ ⎜ 3 .. ⎜ β . 3β 2 3β 1 Tnα := ⎜ ⎜ . .. ⎜ 0 β 3 3β 2 3β ⎜ ⎜ . .. .. .. .. ⎝ .. . . . . 3 3β 2 0 ... 0 β α > 0, β = 0 .. . .. . 4α 27 , and deﬁne the ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∈ Rn×n , ⎟ 0 ⎟ ⎟ ⎟ 1 ⎠ 3β n ∈ N. Note that the eigenvalues of Tnα coincide with the zeros of the monic polynomial α Qα n (z) = det(zIn − Tn ), n ∈ N. These polynomials satisfy the recurrence relation α α 2 α 3 α z Qα n (z) = Qn+1 (z) + 3β Qn (z) + 3β Qn−1 (z) + β Qn−2 (z), (2.15) α α with Qα 0 ≡ 1 and Q−1 ≡ Q−2 ≡ 0. The following asymptotic result for the α eigenvalues of the matrices Tn then follows from Theorem 2.1. Theorem 2.8. The limiting eigenvalue distribution of the matrices Tnα , with α > 0, is given by the measure υ[0,α] , deﬁned as in Theorem 2.1. Proof. The homogeneous recurrence relation α 2 α 3 α 0 = Qα n+1 (0) + 3β Qn (0) + 3β Qn−1 (0) + β Qn−2 (0) α α with Qα 0 (0) = 1 and Q−1 (0) = Q−2 (0) = 0 has the solution 3n n2 n Qα (0) = (−β) 1+ + , n ∈ N. n 2 2 T̃nα 4α 27 > 0 all the Tnα are nonsingular. Next, ⎛ 1 0 ... ... ... ... 0 .. ⎜ .. ⎜ 3β . 1 . ⎜ ⎜ .. . .. ⎜ 3β 2 3β 1 . ⎜ ⎜ . . . . = ⎜ β 3 3β 2 3β . 1 . ⎜ ⎜ . . . ... ⎜ 0 1 β 3 3β 2 3β ⎜ ⎜ . .. .. .. .. .. ⎝ .. . . . . . 0 3 2 3β 3β 1 0 ... 0 β So, since β = deﬁne ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∈ R(n+1)×(n+1) , ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 3 and notice that T̃nα = (Ãα n ) with ⎛ 1 0 ⎜ ⎜ β 1 ⎜ ⎜ α Ãn = ⎜ 0 β ⎜ ⎜ . . .. ⎝ .. 0 ⎞ ... ... 0 .. ⎟ .. . . ⎟ ⎟ .. ⎟ ∈ R(n+1)×(n+1) . .. . 1 . ⎟ ⎟ ⎟ .. . . .. 0 ⎠ ... 0 β 1 α ◦ Since Ãα n is totally nonnegative, T̃n is totally nonnegative, see, e.g., [15, p. 74, 1 ], α α and so also Tn . By [15, p. 100, Theorem 10] we then get that the Tn are oscillation License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS 5579 matrices. Consequently, the zeros of the polynomials Qn are simple and positive [15, p. 87, Theorem 6] and satisfy the interlacing property [15, p. 107, Theorem 14]. The theorem then easily follows from Theorem 2.1 and the recurrence (2.15). Remark 2.9. The polynomials satisfying the recurrence relation with constant coeﬃcients (2.15) are the multiple Chebyshev polynomials of the second kind after a cubic transformation [12, 14]. These are an example of multiple orthogonal polynomials of type II extending the wellknown Chebyshev polynomials of the second kind [13]. The corresponding orthogonality measures can be found in [12, Corollary 4.2], [14, Theorem 4.1]. 3. Ratio asymptotics In [21] Kuijlaars and Van Assche have proven a theorem that gives explicit ratio asymptotics for orthogonal polynomials with converging varying recurrence coeﬃcients. In this section we give an extension of this result to polynomials satisfying a (speciﬁc) fourterm recurrence relation instead of a threeterm recurrence relation. ∞ Theorem 3.1. Suppose we have for each N ∈ N sequences {bn,N }∞ n=0 , {cn,N }n=1 and {dn,N }∞ of real recurrence coeﬃcients and let P be the monic polynomials n,N n=2 generated by the recurrence (3.1) zPn,N (z) = Pn+1,N (z) + bn,N Pn,N (z) + cn,N Pn−1,N (z) + dn,N Pn−2,N (z), with P0,N ≡ 1, P−1,N ≡ 0 and P−2,N ≡ 0. Assume that for some ﬁxed t > 0 the recurrence coeﬃcients have the limits 2 3 4α 4α 4α , lim dn,N = , (3.2) lim bn,N = 3 , lim cn,N = 3 27 27 27 n/N →t n/N →t n/N →t with α ≥ 0. Furthermore, assume that the polynomials Pn,N have real simple zeros xn,N < . . . < xn,N satisfying the interlacing property xn+1,N < xn,N < xn+1,N , n 1 j j j+1 ∗ for all n, N ∈ N, j = 1, . . . , n. Moreover, suppose that for some t > t there exist m ≤ 0, M ≥ α such that all zeros of Pn,N belong to [m, M ] whenever n ≤ t∗ N . Then ⎧ 1 z ⎪ ⎪ α > 0, ⎨ αφ α , Pn,N (z) = (3.3) lim ⎪ n/N →t Pn+1,N (z) ⎪ 1 ⎩ , α = 0, z uniformly on compact subsets of C \ [m, M ], where φ is deﬁned by √ √ 27 3 ω3 1/3 (3.4) φ(z) := z ω3 (−1 + 1 − z)1/3 + (−1 − 1 − z)1/3 − 1 4 2 with ω3 = e (3.5) 2πi 3 and ρeiθ iθ 1/3 (ρe ) iθ = ρ1/2 e 2 , = ρ 1/3 iθ 3 e , ρ > 0, θ ∈ [0, 2π), ρ > 0, θ ∈ (−π, +π]. Remark 3.2. In the case that the recurrence coeﬃcients do not depend on N the existence of the limit (3.3) was already proven in [4]. Our proof will be based on similar arguments. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5580 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE Remark 3.3. Under the conditions of Theorem 3.1, by taking the derivative of (3.3) we get ⎧ 1 φ (z/α) ⎪ ⎪ , α > 0, ⎨ Pn,N (z) Pn+1,N (z) α φ(z/α) − (3.6) lim = ⎪ Pn+1,N (z) n/N →t Pn,N (z) ⎪ ⎩ −1, α = 0, z uniformly on compact subsets of C \ [m, M ]. In order to prove Theorem 3.1 we need part (a) of the following lemma. The whole lemma will be used in the proof of Theorem 2.1 as well. It can be found in, e.g., [21, Lemma 2.2], but we include a short proof for completeness. Lemma 3.4. Suppose that the zeros of the monic polynomials pn−1 and pn , with degree n − 1 and n, respectively, are simple and real, interlace and lie in [m, M ]. Then pn−1 (z) 1 ≤ , ∀z ∈ C \ [m, M ], (a) pn (z) dist(z, [m, M ]) pn−1 (z) ≥ 1 , if z > max(m, M ). (b) pn (z) 2z Proof. Denote the real zeros of pn by y1 , . . . , yn . Since pn−1 and pn are monic and their zeros interlace, there exist wj > 0, nj=1 wj = 1, so that pn−1 (z) wj = . pn (z) z − yj j=1 n Then note that, because yj ∈ [m, M ], for all z ∈ C \ [m, M ] we have z − yj  ≥ dist(z, [m, M ]), 1 ≤ j ≤ n. This immediately proves part (a) of the lemma. If z > max(m, M ), then yj /z < 1 and therefore ( 1−y1j /z ) > 12 , 1 ≤ j ≤ n. Hence ⎛ ⎞ n n n 1 ⎝ 1 wj wj ⎠ 1 1 , ≥ > wj = z j=1 1 − yj /z z 1 − yj /z 2z j=1 2z j=1 which proves part (b). We also need the following properties of the function φ. Lemma 3.5. The function φ is analytic on C \ [0, 1] and satisﬁes 3 4φ(z) , z ∈ C \ [0, 1], (a) zφ(z) = 1 + 27 (b) φ(z) = z −1 + O(z −2 ), as z → ∞. Proof. By the choice of branch cuts for the square and the cubic root, see (3.5), the function φ is certainly analytic on C \ (−∞, 1]. For x < 0, a simple calculation also shows that lim ↓0 φ(x + i) = lim ↓0 φ(x − i). So φ is analytic on C \ [0, 1]. If we deﬁne √ (3.7) u± (z) := −1 ± 1 − z, then (3.8) u+ (z)u− (z) = z, u+ (z) + u− (z) = −2. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS Using this gives 4φ(z) 1+ 27 3 = = 5581 3 27 z ω3 (u+ (z))1/3 + (u− (z))1/3 8 27 z −2 + 3ω3 z 1/3 ω3 (u+ (z))1/3 + (u− (z))1/3 , 8 which veriﬁes part (a) of the lemma. Next, we will prove that φ tends to zero as z → ∞. From part (a) we then obtain that limz→∞ zφ(z) = 1 and, since φ is analytic in a neighborhood of inﬁnity, this implies part (b) of the lemma. Applying the formula (a + b)(a2 + b2 − ab) = a3 + b3 and (3.8), we observe that (3.9) 4 3 ω3 z 1/3 φ(z) = − − 1. 2 2 27 ω 2 (u+ (z))1/3 + (u− (z))1/3 − ω3 z 1/3 3 Now take for a moment z = 1√+ L, with L > 0. By the deﬁnition of the square root we have u± (1 + L) = −1 ± i L. So we can write u+ (1 + L) u− (1 + L) = ρ(L) ei( 2 +ε(L)) , π = ρ(L) ei(− 2 −ε(L)) , Obviously we then get 2 ω32 (u+ (1 + L))1/3 (3.10) 2 (3.11) (u− (1 + L))1/3 π lim ε(L) = 0 and L→+∞ π . 2 = (ρ(L))2/3 ei(− 3 + 3 ε(L)) , π 2 = (ρ(L))2/3 ei(− 3 − 3 ε(L)) . π Finally, notice that (3.12) 0 < ε(L) < ρ(L) ∼ √ L, 2 L → +∞. Hence, combining (3.9), (3.10) and (3.11), we obtain lim L→+∞ 4 3 φ(1 + L) = lim − 1 = 0. L→+∞ ei( 23 ε(L)) + e−i( 23 ε(L)) + 1 27 Since φ is analytic in a neighborhood of inﬁnity, then also limz→∞ φ(z) = 0. Now we give the proof of Theorem 3.1. Proof of Theorem 3.1. It is enough to prove the cases α = 0 and α = 1. The more general case α > 0 is then obtained by taking P̃n,N (z) := Pn,N (αz) /αn . We ﬁrst prove the case α = 1. By the assumptions on the zeros of the polynomials Pn,N every member of Pn,N (z) ∗ (3.13) n, N ∈ N, n ≤ t N Pn+1,N (z) satisﬁes the estimate in part (a) of Lemma 3.4. So, the family (3.13) is uniformly bounded on compact subsets of C \ [m, M ]. By the theorem of Montel [17, p. 563] we then know that (3.13) is a normal family on C \ [m, M ]. For a License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5582 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE sequence {(nj , Nj )}j≥1 , with nj , Nj → ∞, nj /Nj → t as j → ∞, we have that, if j is suﬃciently large, the function (3.14) fj (z) := Pnj ,Nj (z) Pnj +1,Nj (z) belongs to the normal family (3.13). The corresponding sequence {fj }j≥1 then has a subsequence that converges uniformly on compact subsets of C\[m, M ]. If we can prove that the limit of any such subsequence is φ, then, by a standard compactness argument, the full sequence {fj }j≥1 converges uniformly on compact subsets of C \ [m, M ] to φ. This then proves the theorem in the case α = 1. We will show that for each sequence ni , Ni → ∞ with ni /Ni → t such that the functions {fi }i≥1 converge uniformly on compact subsets of C \ [m, M ], we have (3.15) f (z) := lim fi (z) = φ(z) + O(z −k ), i→∞ as z → ∞, for each k ∈ N. The uniqueness of the Laurent expansion around inﬁnity then implies that f (z) = φ(z). We show this by induction on k. The case k = 1 follows from Lemma 3.5 (b) and fi (z) = O(z −1 ), for every i ≥ 1. Next, suppose that the claim holds for some k ≥ 1 and consider a sequence {(ni , Ni )}i≥1 such that ni , Ni → ∞, ni /Ni → x and the functions {fi }i≥1 converge uniformly on compact subsets of C \ [m, M ] to some function f as i → ∞. If we put gi (z) := hi (z) := Pni −1,Ni (z) , Pni ,Ni (z) Pni −2,Ni (z) , Pni −1,Ni (z) z ∈ C \ [m, M ], z ∈ C \ [m, M ], then from the recurrence relation (3.1) we obtain (3.16) z = fi (z)−1 + bni ,Ni + cni ,Ni gi (z) + dni ,Ni hi (z). Since t < t∗ we may assume without loss of generality that ni < t∗ Ni for every i ≥ 1. Then {gi }i≥1 and {hi }i≥1 are subsets of the normal family (3.13). Therefore, there is a sequence ij → ∞, j → ∞, such that {gij }j≥1 and {hij }j≥1 converge uniformly on compact subsets of C \ [m, M ] with limits g and h, respectively. If we pass to such a subsequence and take limits in (3.16), then by (3.2) we ﬁnd 2 3 4 4 1 4 (3.17) z = +3 g(z) + g(z)h(z), z ∈ C \ [m, M ]. +3 f (z) 27 27 27 By the induction hypothesis we now have that g(z) = φ(z) + O(z −k ), h(z) = φ(z) + O(z −k ), z → ∞, z → ∞. Applying this to (3.17), by Lemma 3.5 we then get 2 3 4 4 1 4 1 =z−3 + O(z −k ). φ(z) − φ(z)2 + O(z −k ) = −3 f (z) 27 27 27 φ(z) Since φ(z) = O(z −1 ), this implies f (z) = φ(z) φ(z) = = φ(z) + O(z −k−2 ). 1 + φ(z)O(z −k ) 1 + O(z −k−1 ) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS 5583 So we proved that (3.15) also holds with k replaced by k + 2. Therefore, it holds for all k. Finally, for the case α = 0 the proof is similar. In fact, (3.3) then easily follows by taking limits in (3.16). 4. Proof of Theorem 2.1 In order to prove the asymptotic result in Theorem 2.1 we ﬁrst have a closer look at the function φφ , which is analytic on C \ [0, 1]. Here we will use the relation 4φ(z) −1 +1 3 8φ(z) 1 φ (z) 27 = = + −1 , (4.1) φ(z) 2z 2z 27 z 8φ(z) 27 − 1 which can be obtained by diﬀerentiating part (a) of Lemma 3.5. First of all, we are interested in the jump across its branch cut. Lemma 4.1. The jump of the function (4.2) m(x) := lim ε↓0 φ φ across its branch cut is given by dυ[0,1] φ (x + i) φ (x − iε) − lim = 2πi (x), ε↓0 φ(x − i) φ(x + iε) dx x ∈ (0, 1), where υ[0,1] is deﬁned as in Theorem 2.1. Proof. Let x ∈ (0, 1). By (4.1) we easily obtain −1 8φ(xeiε ) 8φ(xe−iε ) 3 −1 −1 lim − lim (4.3) m(x) = ε↓0 2x ε↓0 27 27 Applying the deﬁnitions (3.5) we get 1/3 lim −1 + 1 − xe±iε ε↓0 1/3 lim −1 − 1 − xe±iε ε↓0 π = ei 3 1 ± √ 1−x = e−i 3 1 ∓ π √ 1/3 1−x −1 . , 1/3 . √ 1 − x)1/3 we then have iπ iπ 8φ(xe±iε ) − 1 = 3x1/3 e− 3 v± (x) + e 3 v∓ (x) − 3. lim ε↓0 27 Using the notation v± (x) := (1 ± So, also applying the relations x1/3 = v+ (x)v− (x) and 2 = v+ (x)3 + v− (x)3 , equation (4.3) becomes √ v+ (x) − v− (x) 2 3i m(x) = 2 2/3 1/3 x x (v+ (x) + v− (x)) − 2 + 3x2/3 (v+ (x) − v− (x))2 √ −1 2 3i = (v+ (x) − v− (x))−1 (v− (x)2 − v+ (x)2 )2 + 3v+ (x)2 v− (x)2 . 2/3 x If we multiply the numerator and denominator both by v+ (x) + v− (x), then we ﬁnally obtain √ √ v+ (x) + v− (x) 3i v+ (x) + v− (x) 2 3i √ m(x) = 2/3 , = 2/3 3 3 3 3 (v+ (x) + v− (x) ) (v+ (x) − v− (x) ) x 2x 1−x which proves (4.2). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5584 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE A second point of interest is the behavior of the function of its branch points 0 and 1. Lemma 4.2. Near the points 0 and 1 we have iθ φ (εe ) −2/3 (4.4) , ε ↓ 0, φ(εeiθ ) = O ε φ (1 + εeiθ ) = O ε−1/2 , ε ↓ 0, (4.5) φ(1 + εeiθ ) φ φ in the neighborhood θ ∈ (0, 2π) , θ ∈ (−π, π) . Proof. For θ ∈ (0, 2π) we easily see that 4φ(εeiθ ) 27 iθ lim φ(εe ) = − , + 1 = O ε1/3 , ε↓0 4 27 ε ↓ 0. Applying this to the ﬁrst equality in (4.1) then gives expression (4.4). √ π+θ We now take θ ∈ (−π, π). We then have u± (1 + εeiθ ) = −1 ± ε ei 2 , where we use the notation (3.7), and applying the deﬁnition of the third root (3.5) we get πi πi 27 27 3ω3 ω3 e 3 + e− 3 − 1 = . (4.6) lim φ(1 + εeiθ ) = ε↓0 4 2 8 If we write u± (1 + εeiθ ) := ρ± (ε, θ)eiη± (ε,θ) , meaning the polar coordinates, then a closer look gives √ η± (ε, θ) = ±π ∓ ε cos (θ/2) + O(ε), ε ↓ 0, and √ i ε cos (θ/2) + O(ε), ε ↓ 0, (4.7) =e 1∓ e 3 √ ε 1/3 ρ± (ε, θ) =1± (4.8) sin (θ/2) + O(ε), ε ↓ 0. 3 From this we easily obtain √ iθ 8φ(1 + εeiθ ) − 1 = − 3εe 2 + O(ε), ε ↓ 0. (4.9) 27 Applying (4.6) and (4.9) to the ﬁrst equality in (4.1) then ﬁnally leads to (4.5). i η± (ε,θ) 3 ± πi 3 As a corollary of Lemma 4.1 and Lemma 4.2 we obtain that transform of the measure υ[0,1] , up to a minus sign. Lemma 4.3. Let φ be deﬁned by (3.4); then 1 φ (z) =− dυ[0,1] (x), (4.10) φ(z) z−x φ φ is the Stieltjes z ∈ C \ [0, 1], with υ[0,1] deﬁned as in Theorem 2.1. Proof. By Lemma 4.1 and Lemma 4.2 and applying Lemma 3.5 (b) to (4.1), the function φφ satisﬁes the following additive RiemannHilbert problem: (P1) f is analytic in C \ [0, 1], (P2) lim f (x + iε) − lim f (x − iε) = 2πi ε↓0 ε↓0 dυ[0,1] dx (x), for x ∈ (0, 1), License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse ZERO DISTRIBUTION OF MULTIPLE ORTHOGONAL POLYNOMIALS (P3) f (z) = −z −1 + O z −2 , 5585 as z → ∞, (P4) f (z) = O z −2/3 , as z → 0, f (z) = O (1 − z)−1/2 , as z → 1. If f and g are both solutions of this RiemannHilbert problem, then it is easily seen that f − g is analytic in C \ {0, 1}. Moreover, 0 and 1 are removable singularities by (P4). Liouville’s Theorem and (P3) then imply f ≡ g, meaning that the RiemannHilbert problem has a unique solution. So it is enough to show that 1 dυ[0,1] (x), z ∈ C \ [0, 1], f (z) := x−z dυ [0,1] satisﬁes (P1)(P4). Properties (P1) and (P3) easily follow from the fact that dx is a probability measure on (0, 1). By the PlemeljSokhotskii formula for Cauchy integrals, see, e.g., [23, p. 43, (18.1)], f satisﬁes (P2). Finally, the behavior at the branch points, see (P4), easily follows from [23, p. 74, (29.5) and (29.6)] and (2.4). Remark 4.4. As an easy consequence of Lemma 4.3 we obtain ⎧ 1 φ (z/α) ⎪ , α > 0, ⎨ 1 α φ(z/α) dυ[0,α] (x) = (4.11) − ⎪ z−x ⎩ −1, α = 0, z where υ[0,α] is deﬁned as in Theorem 2.1. z ∈ C \ [0, α], We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. Let t > 0 and ﬁx a number t > t. Clearly, the convergence (2.2) and the fact that the function α is continuous on [0, ∞) imply that the recurrence coeﬃcients are uniformly bounded if n/N is restricted to compact subsets of [0, ∞). So, (4.12) 0 < R := sup{1 + bn,N  + cn,N  + dn,N  : n ≤ t N } < +∞. By the recurrence (2.1) we have Pn,N (z) = det(zIn − Ln,N ), with ⎛ 1 0 ... ... 0 b0,N .. ⎜ .. ⎜ c1,N b1,N . 1 . ⎜ ⎜ .. . . ⎜ d2,N c2,N b2,N . 1 . ⎜ (4.13) Ln,N = ⎜ . .. ⎜ 0 0 b3,N d3,N c3,N ⎜ ⎜ . . . . . .. .. .. .. ⎝ .. 1 0 ... 0 dn−1,N cn−1,N bn−1,N ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠ As a consequence, the zeros of Pn,N are bounded by Ln ∞ . For n ≤ t N we then have that the simple zeros of Pn,N lie in the interval [−R, R]. Moreover, they are assumed to satisfy the interlacing property (for ﬁxed N ) and one can observe that 3 4α(st) ≤ R, for each 0 < s ≤ 1. α(st) ≤ 1 + 27 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journaltermsofuse 5586 E. COUSSEMENT, J. COUSSEMENT, AND W. VAN ASSCHE So, by Remark 3.3 and Remark 4.4 we establish " ! P sn +1,N (z) P sn ,N (z) 1 − = dυ[0,α(st)] (x) (4.14) lim P sn +1,N (z) P sn ,N (z) z−x n/N →t uniformly on compact subsets of C \ [−R, R], where 0 < s ≤ 1 and sn denotes the greatest integer less than or equal to sn. Note that (4.15) " 1 !P n−1 P sn (z) (z) Pk (z) 1 Pn (z) 1 Pk+1 sn +1 (z) = − − ds. = n Pn (z) n Pk+1 (z) Pk (z) P sn +1 (z) P sn (z) 0 k=0 For n ≤ t N the zeros of the polynomials Pn,N are simple, lie in [−R, R] and satisfy the interlacing property for ﬁxed N . From Lemma 3.4 (b) we then get P sn +1,N (z) ≤ 2z, z > R. P sn ,N (z) With a similar argument as P sn ,N (z) P sn +1,N (z) in Lemma 3.4 (a) we can also prove 1 , z ∈ C \ [−R, R]. ≤ dist(z, [−R, R])2 Combining these two results we have, for z > R, P P sn +1,N (z) P sn ,N (z) sn +1 (z) P sn (z) − = P sn +1 (z) P sn (z) P sn ,N (z) P sn +1,N (z) (4.16) ≤ 2z . dist(z, [−R, R])2 So, we can apply Lebesgue’s dominated convergence theorem on (4.15), and by (4.14) we obtain 1 1 1 dν(Pn,N )(x) = dυ[0,α(st)] (x) ds lim z−x z − x n/N →t 0 1 1 t dυ[0,α(s)] (x)ds, (4.17) = t 0 z−x for z > R. By [16, Theorem 2], which is a gloss on the theorem of Grommer and Hamburger [31, p. 104105], we then ﬁnally establish (2.3). References [1] M. 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