On Nikishin systems with discrete components and weak asymptotics ...

ON NIKISHIN SYSTEMS WITH DISCRETE COMPONENTS AND WEAK ASYMPTOTICS OF MULTIPLE ORTHOGONAL POLYNOMIALS ´ A. I. APTEKAREV, G. LOPEZ LAGOMASINO, AND A. MART´INEZ-FINKELSHTEIN

arXiv:1403.3729v1 [math.CV] 15 Mar 2014

Dedicated to our teachers and friends Andrei Alexandrovich Gonchar and Herbert Stahl

Abstract. We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (σ1 , σ2 ) with unbounded supports (supp σ1 ⊆ R+ , supp σ2 ⊆ R− ) and σ2 is discrete. A Nikishin type equilibrium problem in the presence of an external field acting on R+ and a constraint on R− is stated and solved. The solution is used for deriving the contracted zero distribution of the associated multiple orthogonal polynomials.

Keywords and phrases. Hermite-Pad´e approximants, multiple orthogonal polynomials, discrete orthogonality, weak asymptotic, vector equilibrium problem, Nikishin systems. A.M.S. Subject Classification. Primary: 30E10, 42C05; Secondary: 41A20.

1. Introduction In a celebrated paper published in 1980, E.M. Nikishin [37] introduced a general class of systems of measures, now called Nikishin systems. Let ∆α , ∆β be two non-intersecting bounded intervals of the real line R, and σα ∈ M(∆α ), σβ ∈ M(∆β ), where M(∆) denotes the set of all finite Borel measures on the interval ∆ with constant sign. With σα and σβ we construct a third measure hσα , σβ i, which using the differential notation is given by Z (1) dhσα , σβ i(x) := σ bβ (x)dσα (x), σ bβ (x) = (x − t)−1 dσβ (t). Definition 1.1. Take a collection ∆j , j = 1, . . . , m, of intervals such that ∆j ∩ ∆j+1 = ∅,

j = 1, . . . , m − 1,

and a system of measures (σ1 , . . . , σm ) with σj ∈ M(∆j ), j = 1, . . . , m; we assume additionally that for each j, the convex hull of the support supp(σj ) of σj coincides with ∆j . Let s1 = σ1 , s1 = hσ1 , σ2 i, . . . , sm = hσ1 , hσ2 , . . . , σm ii. The first author received support from RFBR grant 13-01-12430 (OFIm) and the Excellence Chair Program sponsored by Universidad Carlos III de Madrid and the Bank of Santander. The second and the third authors were supported by MICINN of Spain under grants MTM2012-36732-C03-01 and MTM2011-28952-C02-01, respectively, and by the European Regional Development Fund (ERDF). Additionally, the third author was supported by Junta de Andaluc´ıa (the Excellence Grant P09-FQM-4643 and the research group FQM-229) and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almer´ıa. 1

We say that (s1 , . . . , sm ) is the Nikishin system of measures generated by (σ1 , . . . , σm ), and denote it by (s1 , . . . , sm ) = N (σ1 , . . . , σm ). This model system was introduced in order to study general properties of Hermite-Pad´e approximants and multiple orthogonal polynomials . Fix n := (n1 , . . . , nm ) ∈ Zm + \ {0}, where 0 is the m dimensional zero vector. Define Pn as a non-zero polynomial of degree deg(Pn ) ≤ |n| := n1 + · · · + nm such that Z xν Pn (x)dsj (x) = 0, . . . , ν = 0, . . . , nj − 1, j = 1, . . . , m. The existence of Pn reduces to solving a homogeneous linear system of |n| equations on the |n| + 1 coefficients of Pn ; therefore, a non-trivial solution is guaranteed. However, in contrast with the scalar case (m = 1) of standard orthogonal polynomials (OP), uniqueness up to a constant factor is not a trivial matter (and, in general, not true for systems of arbitrary measures (s1 , . . . , sm )). In connection with this question in [37] it was shown that in presence of a Nikishin system uniqueness holds, with deg Pn = |n|, for multi-indices of the form (n + 1, . . . , n + 1, n, . . . , n), and stated without proof that it is also true whenever n1 ≥ · · · ≥ nm . In the sequel we assume that Pn is monic. Motivated by the structure of Nikishin systems, Herbert Stahl studied their analytic and algebraic properties (see [9]). In a series of papers [19]–[21], among other results, K. Driver and H. Stahl showed that uniqueness remains valid whenever nj ≤ nk +1, 1 ≤ k < j ≤ m. The problem for arbitrary multi-indices was definitely solved in [25] (and [26] when the generating measures have unbounded and/or touching supports). A remarkable property of Nikishin orthogonal polynomials is that they not only share orthogonality relations with respect to several measures but they also satisfy full orthogonality relations with respect to a single (varying with respect to n) measure. For m = 2 and n2 ≤ n1 + 1 this was first observed by Andrei Aleksandrovich Gonchar1 by showing that the function of the second kind Z Pn (x) dσ1 (x) Rn,1 (z) = z−x satisfies the orthogonality relations Z (2) xν Rn,1 (x)dσ2 (x) = 0, ν = 0, . . . , n2 − 1. From here it follows that Rn,1 has exactly n2 zeros in C \ ∆1 , they are all simple, and lie in the interior of ∆2 . If Pn,2 denotes the monic polynomial of degree n2 vanishing at these points, then Z dσ1 (x) (3) xν Pn (x) = 0, ν = 0, . . . , n1 + n2 − 1. Pn,2 (x) The study of the asymptotic behavior of multiple orthogonal polynomials is greatly indebted to A. A. Gonchar. In joint papers with E. A. Rakhmanov [27]–[29], they introduced the notion of vector equilibrium problem to describe the asymptotic zero distribution of such polynomials. For a Nikishin system of two measures and n1 = n2 = n the result may be 1On one of the regular Monday seminars at the Steklov Institute A. A. Gonchar was reporting on the

results contained in [37] but after a short while he had to leave because of an important meeting he had to attend. After about an hour he returned and started anew his presentation proving (2) and (3) and from there deduced the convergence of the corresponding Hermite-Pad´e approximants. 2

stated as follows. Define the normalized zero counting measure νP of a polynomial P as νP =

1 deg P

X

δx ,

P (x)=0

where δx denotes the Dirac measure with mass 1 at the point x, and each zero of P is taken with account of its multiplicity, so that the total variation |νP | of νP is 1. Assume that σj ∈ Reg, j = 1, 2 (for the definition of the class Reg of measures, see [50, Chapter 3]). Then there exist positive measures λj ∈ M(∆j ), j = 1, 2, |λ1 | = 2, |λ2 | = 1, such that (4)

lim νPn,2 = λ2 ,

lim νPn = λ1 /2, n

n

in the weak-* topology of measures, where λ1 and λ2 are uniquely determined by the solution of the vector equilibrium problem 2P λ1 (x)

−

P λ2 (x)

−

P λ1 (x)



= γ1 , x ∈ supp(λ1 ), ≥ γ1 , x ∈ ∆1 \ supp(λ1 ),



= γ2 , x ∈ supp(λ2 ), ≥ γ2 , x ∈ ∆2 \ supp(λ2 ),

(5) 2P λ2 (x)

and P λ denotes the logarithmic potential of λ (see the definition below). At the time, this result and its extensions were well known within a small circle of specialists. With some variations, for general Nikishin systems it appeared in papers by H. Stahl [49], and with the highest degree of generality by A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin [30]. For other extensions and generalizations see [5], [7], [12], [15], [24], [38], [42], [43]. In recent years, Nikishin systems have attracted new attention because this construction has been identified in different models of random matrix theory and multiple orthogonal polynomial ensembles, see [6], [32], and [33]. In some of these models new ingredients appear in which some of the generating measures turn out to be discrete and/or have unbounded support. V. N. Sorokin has studied the asymptotic distribution of the zeros for several multiple orthogonal polynomials of this type, see [46]–[47]. Orthogonal polynomials with respect to discrete measures have the characteristic that between two consecutive mass points there may be at most one zero of the polynomial. This fact induces a constraint on the equilibrium problem whose solution describes the asymptotic zero distribution of the orthogonal polynomials. This effect was first pointed out by E. A. Rakhmanov in [41] (see also [18] and [36]). A similar situation occurs in the case of multiple orthogonal polynomials. The present paper is devoted to the study of multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (σ1 , σ2 ) with unbounded supports (supp(σ1 ) ⊆ R+ , supp(σ2 ) ⊆ R− ); moreover, the second measure σ2 is discrete. To obtain the limiting zero distribution (4) of such multiple OP we state and solve a Nikishin type equilibrium problem which generalizes (5) by having an external field acting on R+ and a constraint on R− . The main results are stated in Section 2. In Section 3 we review some examples of explicit solutions of the type of equilibrium problems that we consider. The last two sections include the proofs of the main results. 3

2. Statement of the main results Let dσ1 (x) = σ10 (x)dx be a positive and absolutely continuous measure on R+ , and σ2 a purely discrete measure on R− given by X tk → −∞, λk > 0. (6) σ2 = βk δtk , 0 > t1 > t2 > · · · , k≥1

All the moments of σ1 are assumed to be finite and σ b2 is integrable with respect to σ1 . Let (s1 , s2 ) = N (σ1 , σ2 ) be the Nikishin system generated by these measures. For n = (n1 , n2 ) ∈ Z2+ \ {0} we define Pn as the monic polynomial of degree |n| which satisfies Z (7) xν Pn (x)dsj (x) = 0, ν = 0, . . . , nj − 1, j = 1, 2. The zeros of Pn are simple, lie in the interior of R+ , and deg Pn = n1 + n2 . We will restrict our attention to sequences of multi-indices of the form n = (n, n). In order to simplify the notation we write Pn instead of Pn . Thus, deg Pn = 2n. Our goal is to describe the (rescaled) asymptotic zero distribution of the polynomials (Pn ) , n ∈ N, under appropriate assumptions on the generating measures σj , j = 1, 2. Using the properties of Nikishin systems (see [26] and [30]) it is easy to deduce that there exists a monic polynomial Pn,2 , deg Pn,2 = n, whose zeros are simple and contained in the interior of the convex hull of supp(σ2 ), such that Z Pn (x) (8) xν dσ1 (x) = 0, ν = 0, . . . , 2n − 1, Pn,2 (x) and Z (9)

ν Pn,2 (t)

t

Pn (t)

Z

Pn2 (x) dσ1 (x) dσ2 (t) = 0, Pn,2 (x) x − t

ν = 0, . . . , n − 1.

In other words, Pn and Pn,2 satisfy full orthogonality relations with respect to varying measures. Let (dn )n∈Z+ , dn ≥ 1, be a sequence of numbers, and let Qn (x) = Pn (dn x)/d2n n ,

(10)

Qn,2 (t) = Pn,2 (dn t)/dnn .

Making the change of variables x → dn x, t → dn t it follows that the monic polynomials Qn , Qn,2 verify the orthogonality relations Z Qn (x) 0 σ (dn x)dx = 0, ν = 0, . . . , 2n − 1, (11) xν Qn,2 (x) 1 and Z (12)

t

ν Qn,2 (t)

Qn (t)

Z

Q2n (x) σ10 (dn x)dx dσ2,n (t) = 0, Qn,2 (x) x − t

ν = 0, . . . , n − 1,

where (13)

dσ2,n (t) =

X

βk δξk,n (t),

ξk,n = tk /dn .

k≥1

We must impose some restrictions on the points ξk,n and the numbers βk , dn : (i) There exists a positive continuous function ρ on R− such that for every compact set K ⊂ R− , |ξk+1,n − ξk,n | > ρ(ξk,n )/n, k ≥ 1. 4

(ii) There exist two positive functions A(x), B(n), such that #{k : ξk,n ∈ [x, 0]} ≤ A(x)B(n),

x < 0,

where A and B satisfy that for every β > 0, lim

x→−∞

log A(x) =0 |x|β

and

lim n

log B(n) = 0. n

(iii) For each fixed x < 0 lim (inf{βk : ξk,n ∈ [x, 0]})1/n = 1. n

(iv) There exists a finite positive Borel measures σ supported on R− , |σ| > 1, such that for every compact subset K ⊂ R− , the logarithmic potential P σ|K (see the definition in (17)) of the restriction of σ to K is continuous on C and Z Z 1 X f (x)δξk,n = f (x)dσ(x) (14) lim n→∞ n k≥0

for every bounded continuous function f with compact support in R− . (v) There exists a continuous function ϕ on R+ such that 1 (15) lim log σ10 (dn x) = −ϕ(x) n→∞ n uniformly on each compact subset of R+ , for which the following condition holds: (16)

lim (ϕ(x) − 4 log x) = +∞.

x→+∞

In order to describe the zero asymptotic behaviour of the multiple orthogonal polynomials Qn , Qn,2 , we need to solve an associated vector equilibrium problem that we now present. For a closed subset ∆ ⊂ R we denote by M+ (∆) the class of all finite positive Borel measures µ such that supp(µ) ⊂ ∆. We write µ ∈ M+ c (∆) if, additionally, |µ| = c. Let µ ∈ M+ (R). Its logarithmic potential and energy are defined as Z Z Z 1 1 (17) P µ (x) := log dµ(y), I(µ) := log dµ(x)dµ(y), |x − y| |x − y| respectively, whenever these integrals are well defined. Assume that µ1 , µ2 ∈ M+ (R) verify Z (18) I(µ) < +∞, log(1 + |x|2 )dµ(x) < +∞. Their mutual energy may be defined as Z Z I(µ1 , µ2 ) := log

1 dµ1 (x)dµ2 (y). |x − y|

Analogously, one can define the potential, energy, and mutual energy of signed measures. In particular, if (18) takes place then I(µ1 − µ2 ) = I(µ1 ) + I(µ2 ) − 2I(µ1 , µ2 ). Moreover, for µ1 , µ2 ∈ M+ c (R), we have (19)

I(µ1 − µ2 ) ≥ 0,

with equality if and only if µ1 = µ2 (see [45], also [48] if the measures have a bounded support). 5

As in (iv) above, let σ ∈ M+ (R− ), supp(σ) = R− , |σ| > 1, be such that for every compact subset K ⊂ R− we have that P σ|K is continuous on C (recall that σ|K denotes the restriction of σ to K). We define (20)

+ M(σ) := {~ µ = (µ1 , µ2 )t ∈ M+ 2 (R+ ) × M1 (R− ) : µ2 ≤ σ},

where the superscript t stands for transpose. By µ2 ≤ σ we mean that σ − µ2 is a positive measure. Since we have assumed that P σ|K is continuous on C for every compact K, it readily follows that P µ2 is continuous on C. Let ϕ be a continuous function on R+ satisfying (16). Unless otherwise stated, in what follows we assume that σ and ϕ verify all the properties just described. Set Z Z ∗ 2 M (σ) := {~ µ ∈ M(σ) : log(1 + y )dµ1 (y) < +∞, log(1 + y 2 )dµ2 (y) < +∞},   Z Jϕ (~ µ) = 2 I(µ1 ) − I(µ1 , µ2 ) + I(µ2 ) + ϕ dµ1 ,

∗

Jϕ := inf{Jϕ (~ µ) : µ ~ ∈ M (σ)}, and

W1µ~ (x) := 2P µ1 (x) − P µ2 (x) + ϕ(x),

~

W2λ (x) := 2P λ2 (x) − P λ1 (x).

Theorem 2.1. With the assumptions on ϕ and σ above, the following statements are equivalent and have the same unique solution: (A) There exists ~λ ∈ M∗ (σ) such that Jϕ (~λ) = Jϕ . (B) There exists ~λ ∈ M∗ (σ) such that for all ~ν ∈ M(σ), Z Z ~λ ~ W1 d(ν1 − λ1 ) + W2λ d(ν2 − λ2 ) ≥ 0. (C) There exist ~λ = (λ1 , λ2 ) ∈ M∗ (σ) and constants w1 , w2 such that (21)

(22)

λ1



λ2

2P (x) − P (x) + ϕ(x) λ2

λ1

2P (x) − P (x)



= w1 , x ∈ supp(λ1 ), ≥ w1 , x ∈ R + ,

≤ w2 , x ∈ supp(λ2 ), ≥ w2 , x ∈ supp(σ − λ2 ).

The constants w1 , w2 are uniquely determined as well. We also have that P λ1 , P λ2 are continuous on C, supp(λ1 ) is compact, and supp(λ2 ) is connected. If xϕ0 (x) > 0 is increasing on R+ then supp(λ1 ) is also connected. Should supp(λ2 ) ∩ supp(σ − λ2 ) be unbounded, we have w2 = 0. Results of this nature (in a more general setting regarding the dimension of the vector equilibrium problem and the supports of the corresponding measures) may be seen in [11]. There, the action of constraints on the measures is not considered and the external fields, which verify restrictions of the form (16), act on all the components of the vector measures. This implies in turn that all the components of the equilibrium vector measure have compact support. However, taking into consideration certain applications, we are especially interested in allowing the second component of the equilibrium measure to be unbounded. For this reason, in the proof of Theorem 2.1 (see also Lemma 4.1) we follow the approach presented in [31] where results similar to Theorem 2.1, except for part (C), also appear. We wish to point out that Theorem 2.1 remains valid if supp(σ) is any non-trivial closed interval contained in R− . 6

Now we are ready to formulate the main result about the zero asymptotics of Nikishin orthogonal polynomial. Theorem 2.2. Let the assumptions (i)–(v) formulated above hold, and let ~λ = (λ1 , λ2 ) ∈ M∗ (σ) be the solution of the extremal problem in Theorem 2.1. Then lim νQn,2 = λ2 ,

lim νQn = λ1 /2, n

n

in the weak-* topology of measures. Also Z 1/n 2 (x) Q n 0 (23) lim σ1 (dn x)dx = e−w1 , n Qn,2 (x) and (24)

1/n Z Q2 (t) Z Q2 (x) σ 0 (d x)dx n,2 n n 1 dσ2,n (t) = e−(w1 +w2 ) , lim n Qn (t) Qn,2 (x) x − t

where w1 , w2 are the corresponding equilibrium constants from (21)–(22). Although the assumptions of this theorem may seem too restrictive, it encompasses many interesting examples. Some of them will be discussed in the next Section. In particular, we will analyze briefly the case of the modified Bessel weights (appearing in the analysis of the non-intersecting squared Bessel paths), the multiple Hermite polynomials (useful when studying ensembles of random matrices with an external source), and finally, the multiple Pollaczek polynomials, studied previously in [46], which will be discussed in more detail, and for which an alternative method for solving the equilibrium problem of Theorem 2.1 is presented. Let us finish this section noting that we can easily translate the results of Theorems 2.1 and 2.2 to the equivalent setting of the whole real axis R (with symmetric measures with respect to the origin). Indeed, let {Pm } be a sequence of multiple orthogonal polynomials satisfying (7) with respect to a Nikishin system (8)–(9) on the semiaxis R+ , and define the polynomial sequence {P˜n } with polynomials of even degrees by n (25) P˜n (x) := Pm (x2 ), m = , n ∈ 2N. 2 ˜ Then Pn are multiple orthogonal polynomials satisfying conditions of the form (7) with respect to what can be seen as a natural generalization of a Nikishin system: now the first generating measure σ1 is supported on the whole real axis R, while the second generating measure σ2 is a discrete measure on the imaginary axis. Then for the rescaled polynomials ˜ n (x) := Pm (dm x2 )/d2m we have straightforward analogues of Theorems 2.1 and 2.2, but Q m now in terms of the solution of the following equilibrium problem: there exists a unique pair of measures (λ1 , λ2 ), |λ1 | = 2, |λ2 | = 1, and unique constants γ1 , γ2 , such that λ2 (x) ≤ σ ˜ for x ∈ iR,  = γ1 , x ∈ supp(λ1 ) ⊂ R, λ1 λ2 (26) 2P (x) − P (x) + ϕ˜ ≥ γ1 , x ∈ R, (27)

λ2

λ1

2P (x) − P (x)



≤ γ2 , x ∈ supp(λ2 ) ⊂ iR, ≥ γ2 , x ∈ supp(σ − λ2 ).

The external field and the constraint are related to their analogues in (21)–(22) by ϕ(x) ˜ = ϕ(x2 ), σ ˜ (x) = σ(x2 ). 7

3. Examples of explicit solutions of the equilibrium problem As we already mentioned in the introduction, in recent years various models from random matrix theory have been reformulated in terms of multiple orthogonal polynomials corresponding to Nikishin systems of type (7)–(9). In all of them, the generated weights are given by entire functions whose ratio is a meromorphic function, which can be considered as the Cauchy transform of a discrete measure σ2 as in (6). In this section we discuss three examples of this type of Nikishin systems for which explicit solutions of the associated equilibrium problems stated in Theorem 2.1 are available. One of them (see subsection 3.3 below) is analyzed in more detail, along with a new approach for expressing the density of the equilibrium measure as a jump of the logarithm of an algebraic function. In this representation the constrained (by the Lebesgue mesure) part of the equilibrium measure is modeled as the jump of the logarithm of a negative function. In contrast to the standard approach, where either the underlying differential equations or the recurrence relations of the corresponding multiple orthogonal polynomials are used, we derive this representation directly from the equilibrium conditions. 3.1. Modified Bessel weights (and non-intersecting squared Bessel paths). In [22], [23] multiple orthogonal polynomials {Pn } satisfying (7) for the system of the weights √
x x , s01 (x) = xα/2 e− 2 Iα x ∈ R+ , √
x s02 (x) = x(α+1)/2 e− 2 Iα+1 x , where Iα is the modified Bessel function, α > −1, were introduced and studied. This system has found applications in the description of ensembles of particles following non-intersecting squared Bessel paths [33], [34]. The polynomials {Pn }, rescaled as in (10), allow for an application of the general Theorem 2.2. Therefore, their weak asymptotics is described by means of the extremal problem solved in Theorem 2.1, with a particular choice of the external field ϕ and the upper constraint σ; namely, p |x| dσ x √ (28) ϕ(x) = − x, x > 0, = , x < 0. 2 dx π An explicit solution of the equilibrium problem (21)–(22) and (28) is known (see [33], or [6], page 1188). The measures λj , j = 1, 2, are absolutely continuous with respect to the Lebesgue measure with densities that can be expressed in terms of solutions of the cubic equation (a.k.a. the spectral curve) (29)

H 3 − 2H 2 + H −

2 = 0. z

Equation (29) has three solutions, enumerated in such a way that 2 + O(z −2 ), z √ 2 1 H1 (z) = 1 − 1/2 − + O(z −3/2 ), z z√ 2 1 H2 (z) = 1 + 1/2 − + O(z −3/2 ), z z

H0 (z) =

8

as z → ∞. Then, as it was shown in [33], λ1 and λ2 can be written as 1 Im H0,+ (x), x > 0, π (30) 1 dσ − Im H1,+ (x), x < 0, λ02 (x) = dx π where the + subindices indicate the boundary values from the upper half plane. λ01 (x) =

3.2. Multiple Hermite polynomials (and random matrices with an external source). Another set of multiple orthogonal polynomials was described in [4]. It turns out that it is more convenient to deal with the polynomials {P˜n }, defined by (25), with respect to the system of the weights 1 2 −aj x)

s0j (x) = e−n( 2 x

x ∈ R,

,

j = 1, . . . , p.

This system has found applications in the description of ensembles of non-intersecting Brownian bridges or random matrices with external source [3], [14]. There, for the case p = 2 and ˜ n} a1 = −a2 = a, it was proved that the zero counting measures of the scaled polynomials {Q (corresponding to {P˜n }) have a weak limit λ which can be described by means of the spectral curve H 3 − zH 2 + (2 − a2 )H + za2 = 0.

(31)

This equation is due to Pastur [40]. If we enumerate the branches in (31) so that, as z → ∞, 2 + O(z −2 ), z 1 H1 (z) = a + + O(z −2 ), z 1 H2 (z) = −a + + O(z −2 ), z H0 (z) = z −

then λ is given by 1 Im H0,+ (x), x ∈ R. π It was noticed in [13] that the measure λ in (32) coincides with the component λ1 in the solution of the equilibrium problem (26)–(27) corresponding to the external field ϕ˜ and the constraint σ ˜ as follows: λ0 (x) =

(32)

ϕ(x) ˜ =

x2 − a|x|, 2

x ∈ R,

de σ (z) =

a |dz|, π

z ∈ iR.

Actually, [13] contains a more general result for the multiple orthogonal polynomials {P˜n } given by (25), corresponding to the system of weights s0j (x) = e−(V (x)−aj x) ,

x ∈ R,

Pd

j = 1, 2,

where V (x) = j=1 vj x2d is an even polynomial potential with vd > 0; it was shown that ˜ n } (corresponding to {P˜n }) converge the zero counting measures of the scaled polynomials {Q (in a weak-* sense) to the first component λ = λ1 of the solution to the equilibrium problem (26)–(27), with the external field ϕ˜ and the constraint σ ˜ given by a (33) ϕ(x) ˜ = V (x) − a|x|, x ∈ R, de σ (z) = |dz|, z ∈ iR. π 9

Moreover, it was also proved in [13] that the equilibrium problem (26)–(27) with input data (33) has always a unique solution (λ1 , λ2 ), |λ1 | = 2, |λ2 | = 1, and that the functions Z dλ1 (s) 0 H0 (z) = V (z) − , z ∈ C \ S(λ1 ), z−s Z Z dλ2 (s) dλ1 (s) − , z ∈ C \ (S(λ1 ) ∪ S(σ − λ2 )) , ± Re z > 0, H1 (z) = ±a + z−s z−s Z dλ2 (s) H2 (z) = ∓a + , z ∈ C \ S(σ − λ2 ), ± Re z > 0, z−s are the three solutions of the equation H 3 + p2 (z)H 2 + p1 (z)H + p0 (z) = 0

(34)

with polynomial coefficients, whose degrees can be easily determined from the degree of the potential V . However, finding the coefficients of these polynomials explicitly in the most general situation is a very difficult problem. In [8] (see also [13]) this was done for a general even quartic potential, b 1 V (x) = x4 − x2 4 2 in the cases when the Riemann surface of (34) is of genus either 0 or 1. For instance, when the genus is 1 we have from [8] that H 3 − (z 3 + bz)H 2 + z 2 H + a2 z 3 = 0, where a and b belong to the triangular domain on the (a, b)-plane, bounded by the curves p √ 6b3 − 27b − 6(b2 − 3)3/2 am (b) := > 0, b ∈ (−2, − 3), 9 p 3 √ 6b − 27b + 6(b2 − 3)3/2 aM (b) := > 0, b ∈ (−∞, − 3). 9 and by the b-axis (a = 0). 3.3. Multiple Pollaczek polynomials. We have come to the main example of the application of the general Theorems 2.2 and 2.1 and of their analogues for the real axis R, as discussed at the end of Section 2. The sequence of polynomials, studied in [46], is defined by the multiple orthogonality conditions (7) on R+ with √

(35)

ds1 (x) =

dx

√

sinh π 2 x

,

ds2 (x) =

1

√

cosh π 2 x

tanh π 2 x dx √ = √ ds1 (x). x x

Decomposing tanh(πz/2)/z into simple fractions, it is easy to check that √ Z tanh π 2 z 4X 1 dσ2 (x) √ = = 2 π z + (2k + 1) z−x z k≥0

where σ2 =

4 X δ−(2k+1)2 π k∈Z+

(cf. (6)). Hence, (s1 , s2 ) = N (σ1 , σ2 ) is a Nikishin system generated by σ1 = s1 , supported on R+ , and the discrete measure σ2 made of equal masses of size 4/π, whose support is contained in (−∞, 0). In this case, the re-scaling (10) is done taking dn = 4n2 . This yields the measure 10

σ2,n , see (13), with ξk,n = −((2k + 1)/2n)2 and βk = 4/π. It is easy to check that conditions (i)–(v) of Section 2 are satisfied with p p √ (36) ρ(x) = A(x) = |x|, dσ(x) = dx/ |x|, ϕ(x) = π x, B(n) = n, so that Theorem 2.2 can be applied. Obviously, a pair of measures (f ds1 , f ds2 ), where f is any bounded positive continuous function on R+ , has associated the same vector equilibrium problem. Thus, the corresponding multiple orthogonal polynomials exhibit the same rescaled normalized zero distribution as those corresponding to (35). Other examples may be constructed replacing the discrete component of the Nikishin system by a Meixner or a Charlier type measure (see, for example, [36], [47] or [2]). We will also consider the corresponding polynomials transplanted to the whole real axis, for multi-indices of the form (n, n). Using the transformation (25) we obtain a sequence of monic polynomials P˜n of degree 2n, satisfying the orthogonality relations Z xdx = 0, ν = 0, . . . , n − 1, xν P˜n (x) (37) sinh πx R Z dx (38) xν P˜n (x) = 0, ν = 0, . . . , n − 1, cosh πx R that are known as multiple (or generalized) Pollaczek polynomials (see [46]). In order to guarantee normality, we will assume additionally that the n are even. In this case, the zeros of P˜n are real and simple. In a similar fashion as it is done for Nikishin systems (on the real line) it can be deduced that there exists a monic polynomial P˜n,2 , deg P˜n,2 = n, whose zeros are also simple and contained in iR \ {0}, such that Z xdx P˜n (x) = 0, ν = 0, . . . , 2n − 1, (39) xν ˜ Pn,2 (x) sinh(πx) R and Z ˜ P˜n2 (x) xdx t dβ(t) = 0, P˜n (t) iR P˜n,2 (x) (x − t) sinh(πx) R

Z (40)

ν Pn,2 (t)

ν = 0, . . . , n − 1.

Set ˜ n (z) = P˜n (nz)/n2n , ˜ n,2 (z) = P˜n,2 (nz)/nn . Q Q The logarithmic (weak) asymptotic behavior of these polynomials was studied by V. N. Sorokin in [46]. Sorokin’s approach is based on the existence of an explicit expression of the gener˜ n (x), to which a weak form of the Darboux method can ating function for the polynomials Q be applied. On this path, the weak asymptotics of the polynomials can be deduced from the singularities of the generating function. ˜ n } (corresponding to By (36), the zero counting measures of the scaled polynomials {Q ˜ {Pn }) have a weak limit λ, which is the first component (λ = λ1 ) of the solution to the equilibrium problem (26)–(27), with (41)

ϕ(x) ˜ = π|x|,

x ∈ R,

de σ (z) = |dz|

on iR.

One of the goals of this section is to obtain λ by a direct solution of this equilibrium problem. From electrostatic considerations we expect that supp(λ2 ) = iR, because the external field created by P λ1 on iR is too weak to make supp(λ2 ) compact. An alternative argument is that, if there were no restrictions on λ2 , the measure 2λ2 in (27) would coincide with the balayage 11

of λ1 onto iR. Hence, the upper constraint forces the balayage measure to redistribute its mass precisely where it exceeds σ in order to attain equilibrium on the rest of iR. This consideration makes us look for a solution λ2 for which there is an equality on supp(σ − λ2 ) in the equilibrium conditions (27). We shall try to find the Cauchy transform of the equilibrium measure λ1 , Z dλ1 (x) b1 (z) = (42) H(z) := −λ . R x−z If we “complexify” the equilibrium relations (26)–(27) and (41), differentiate them and take the real parts, we obtain    −π, on R− ∩ supp(λ1 ), b b Re 2λ1 (x) − λ2 (x) = π, on R+ ∩ supp(λ1 ), and

  b2 (x) − λ b1 (x) = 0, on supp(σ − λ2 ). Re 2λ

Using the Riemann–Schwartz symmetry principle, from the first relation we deduce that the function H can be continued analytically from both sides of the cut along R− ∩ supp(λ1 ). Thus, H can be lifted to a Riemann surface, where (43)

b1 (z) − λ b2 (z) := H1 (z) H(z) = π + λ

is considered on the next sheet. Analogously, H can be continued analytically from both sides of the cut along R+ ∩ supp(λ1 ), so that (44)

b1 (z) − λ b2 (z) := H2 (z) H(z) = −π + λ

is defined on another sheet of the same surface. Let us assume that the complete Riemann surface R = {R(j) }2j=0 , R(j) = C, has three sheets. With appropriate cuts we will have b1 (z) is holomorphic in C \ supp(λ1 ), and three branches of H = {Hj }2j=0 , where H0 (z) = −λ (42)–(44) give us that, as z → ∞, 2 H0 (z) = − + . . . z 1 (45) H1 (z) = π + + . . . z 1 H2 (z) = −π + + . . . . z We make an ansatz that the function H can be found in the form 2 (46) H(ζ) = log ψ(ζ) on R \ {ζ ∈ R : ψ(ζ) ∈ R− }, i where ψ is a meromorphic function on the compact three sheeted Riemann surface R. At this moment, R is still unknown (should it exist); however, representation (46) and relations (45) yield that   1 − ζi + . . . , ζ → ∞(0) ,    1 (47) ψ(ζ) = i − 2ζ + ..., ζ → ∞(1) ,    −i + 1 + . . . , ζ → ∞(2) , 2ζ where q (j) denotes the point on R(j) whose canonical projection on the plane is q ∈ C. We try to take ψ as the simplest meromorphic function which maps R conformally onto C. The 12

inverse of this function is a rational function ζ = r(ψ). From the main term in the asymptotic expansion (47) we have that A B A + + , ψ−1 ψ−i ψ+i

ζ= and the second term gives us that

A = −i,

B=

−1 , 2

1 C= . 2

Thus, ζ = −i

(48)

(ψ 2

ψ(ψ + 1) + 1)(ψ − 1)

or, what is the same, (49)

ψ3 +

i−ζ 2 i+ζ ψ + ψ − 1 = 0. ζ ζ

The discriminant of (49) is equal to 16ζ 4 − 44ζ 2 − 1. Therefore, the algebraic fuunction has four branch points ±e1 and ±e2 , where q q √ √ i 1 22 − 10 5, e2 = −22 + 10 5. e1 = 4 4 Taking into account (47) we fix the following sheet structure of R (see Figure 1) (50) R(0) := C \ [−e1 , e1 ], R(1) := C \ ([−e1 , 0] ∪ [−e2 , e2 ]), R(2) := C \ ([0, e1 ] ∪ [−e2 , e2 ]).

−e1

e1

−e1

R(1)

0

e1

0

R(0)

R(2)

Figure 1. Sheet structure of the Riemann surface R. Therefore, the algebraic function ψ has the following single-valued meromorphic branches (in fact holomorphic, since ψ(0) = {0, −1, ∞}): ψ0 (ζ) ∈ H(C \ [−e1 , e1 ]),

ψ1 (ζ) ∈ H(C \ ([−e1 , 0] ∪ [−e2 , e2 ])),

ψ2 (ζ) ∈ H(C \ ([0, e1 ] ∪ [−e2 .e2 ])), where H(Ω) stands for the class of functions holomorphic (and single-valued) in a domain Ω. From the analysis of the roots of (49) it follows that (51) {iR}(0) = {ζ ∈ R : ψ(ζ) ∈ R+ },

{[−e2 , e2 ]}(1) ∪{[−e2 , e2 ]}(2) = {ζ ∈ R : ψ(ζ) ∈ R− }. 13

Thus, if we cut our compact Riemann surface R along the second set in (51) and denote e := R \ ({[−e2 , e2 ]}(1) ∪ {[−e2 , e2 ]}(2) ), R

(52)

we get that the function H in (46) is single-valued and holomorphic in the open Riemann e Now, we can formulate our result about the solution of the equilibrium problem surface R. (26)–(27): Proposition 3.1. Let 2 log ψj (ζ), ζ ∈ R(j) , j = 0, 1, i where the ψj are the solutions of (49) satisfying (47). Define the absolutely continuous measures dλ1 (x) = λ01 (x)dx, dλ2 (x) = λ02 (x)|dx|, by 1 lim | Im H0 (x + iε)|, x ∈ R, λ01 (x) = π ε→0+ (53) 1 λ02 (x) = −1 + lim Re H1 (x − ε), x ∈ iR = supp(λ2 ). π ε→0+ The pair (λ1 , λ2 ) is the solution of the equilibrium problem (26)–(27) and (41). More precisely, |λ1 | = 2, |λ2 | = 1, and these measures verify Hj (ζ) =

(54) (55)

λ2 ≤ σ, and λ02 (x) = 1 for x ∈ [−e2 , e2 ];  = γ1 , x ∈ [−e1 , e1 ] = supp(λ1 ) ⊂ R, λ1 λ2 2P (x) − P (x) + π|x| > γ1 , x ∈ R, for dσ(x) = |dx|,

and (56)

λ2

λ1



2P (x) − P (x)

= γ2 , x ∈ supp(σ − λ2 ) = iR \ [−e2 , e2 ], < γ2 , x ∈ [−e2 , e2 ].

Before proving Proposition 3.1 we discuss some properties of the primitive function G defined by G0 = H,

(57)

e That is which we now consider on the open Riemann surface, R. Z ζ e (58) G(ζ) = H(t)dt, ζ0 , ζ, t ∈ R. ζ0

The uniformization of R defined in (48) allows us to integrate by parts obtaining Z ψ(ζ) ψ(ψ + 1) = C+ζH(ζ)+2 log(ψ(ζ)−1)−log(ψ 2 (ζ)+1), (59) G(ζ) = −2 log(ψ) d 2 (ψ + 1)(ψ − 1) ψ(ζ0 ) e and where C is a constant which depends on ζ0 . According to (59), G is multivalued on R has local analytic extension to the whole R (and beyond), with possible singular points at ζ = 0 and ζ = ∞ (notice that by (48), ψ(∞) = {1, i, −i}). However, its periods are purely imaginary. Therefore, its real part is a single valued harmonic function on R \ {0, ∞}, g := {gj = Re Gj }2j=0 , which is defined up to an additive constant. We fix the constant so that g0 (∞) + g1 (∞) + g3 (∞) = 0. 14

This normalization in turn implies that (60)

g0 (ζ) + g1 (ζ) + g2 (ζ) ≡ 0,

ζ ∈ C.

Indeed, g0 + g1 + g2 is a symmetric function of g which is harmonic on C \ {0, ∞}. From (48) and (59), one sees that the singularity it has at ζ = 0 is removable. On the other hand, from (45) and (58), we have that the branches of g at infinity have the following behavior   ζ → ∞(0) , −2 log |ζ|, (61) g(ζ) ' π Re z + log |ζ|, ζ → ∞(1) ,   −π Re z + log |ζ|, ζ → ∞(2) . So, ζ = ∞ is also a removable singularity of g0 + g1 + g2 . Since g0 + g1 + g2 is harmonic in C and equal to zero at ∞, it is identically equal to zero. Proof of Proposition 3.1. We must verify that the measures defined by their densities in (53) verify (54)–(56). In order to identify the potentials of the measures λ1 , λ2 , let us change the sheet structure of R. Define ( ( g (z), Re z < 0, g2 (z), Re z < 0, 1 (62) g0∗ := g0 , g1∗ := g2∗ := g2 (z), Re z > 0, g1 (z), Re z > 0. On iR, g ∗ is defined by continuity. Notice that now g1∗ , g2∗ have a harmonic continuation through the interval [−e2 , e2 ]. Now, we see that the function g0∗ is superharmonic, and that g2∗ is subharmonic (being the maximum of two harmonic functions). Therefore, taking into account the behavior at ∞ (see (61)), from the Riesz decomposition theorem for superharmonic functions we obtain a global representation of the branches of g ∗ in C in the form (63)

g0∗ (z) = P λ1 (z) + κ1 , g2∗ (z) = −P λ2 (z) − v(z) + κ2 ,

where λ1 , λ2 are measures supported on [−e1 , e1 ] and iR, respectively, and v(z) is the superharmonic function  π Re z, Re z ≤ 0, (64) v(z) = −π Re z, − Re z > 0. As a consequence of (60), we also have that (65)

g1∗ (z) = −P λ1 (z) + P λ2 (z) + v(z) − κ1 − κ2 .

Using (45) and (63), it is easy to verify that |λ1 | = 2,

|λ2 | = 1,

and taking into consideration the definition of g, the Stieltjes-Perron formula applied to the calculation of the measures yields (53). Since g0∗ (x) = g1∗ (x) for x ∈ [−e1 , e1 ], using (63) and (65) we obtain the equality in (55) with γ1 := −2κ1 − κ2 . The fact that g0∗ (x) > g1∗ (x) on R \ [−e1 , e1 ] allows us to verify the inequality in (55). Analogously, comparing g1∗ and g2∗ on iR, and using (63), (65) and the fact that v(z) ≡ 0, z ∈ iR (see (64)), we obtain (56) with γ2 := 2κ1 + κ2 . Finally, notice that the functions ψ1 , ψ2 have negative limiting values on [−e2 , e2 ] (see the second relation in (51)). Therefore, taking into consideration (46), it follows that lim Re H1 (x − ε) = 2π,

ε→0+

15

x ∈ [−e2 .e2 ],

and λ02 (x) ≡ 1, x ∈ [−e2 , e2 ]. On the rest of the imaginary axis, π < lim Re H1 (x − ε) < 2π ε→0+

(see also (45)). Thus, we obtain (54). We wish to remark that when applying the StieltjesPerrron formula in the second half of (53) we take the imaginary part because |dx| = −idx, x ∈ iR. This concludes the proof.  4. Proof of Theorem 2.1 In order to deal with the condition (18), not given a priori, we will employ the approach presented in [31]. For arbitrary µ1 , µ2 ∈ M+ (R), we define a modified logarithmic potential and mutual energy as follows p Z 1 + y2 µ1 dµ1 (y), V (x) := log |x − y| p √ Z Z 1 + x2 1 + y 2 I(µ1 , µ2 ) := log dµ1 (y)dµ2 (x). |x − y| The modified energy of µ is then given by I(µ) := I(µ, µ). The new kernel is connected with the inverse stereographic projection from the ball in R3 centered at (0, 0, 1/2) and radius 1/2 onto the extended complex plane. Therefore, p √ 1 + x2 1 + y 2 ≥1 (66) |x − y| (for more details see (2.9)–(2.11) in [31]). Consequently, the modified potential and the mutual energy are uniformly bounded from below for all µ1 , µ2 ∈ M+ (R). In case that µ1 , µ2 fulfill (18), Z Z |µ2 | |µ1 | I(µ1 , µ2 ) = I(µ1 , µ2 ) + log(1 + x2 )dµ1 (x) + log(1 + x2 )dµ2 (x). 2 2 Let ϕ be a contiuous function on R+ which verifies
(67) lim inf 2ϕ(x) − 3 log(1 + x2 ) > −∞. x→+∞

This assumption is much weaker than (16). Set ϕ∗ (x) := ϕ(x) − 32 log(1 + x2 ), and define    ∗ 2 −1 ϕ A= , f= . −1 2 0 For µ ~ = (µ1 , µ2 )t ∈ M(σ) (see the definition in (20)), we introduce the vector function p √ Z 1 + x2 1 + y 2 µ ~ µ ~ µ ~ t W (x) = (W1 (x), W2 (x)) := log dA~ µ(y) + f (x) |x − y| and the functional Z Z Z µ ~ µ ~ ∗ (68) Jϕ∗ (~ µ) := (W (x) + f (x)) d~ µ(x) = (W1 (x) + ϕ (x))dµ1 (x) + W2µ~ (x)dµ2 (x) (when either I(µ1 ) = +∞ or I(µ2 ) = +∞, we take Jϕ∗ (~ µ) = +∞). That is, Z Jϕ∗ (~ µ) = 2(I(µ1 ) − I(µ1 , µ2 ) + I(µ2 )) + (2ϕ − 3 log(1 + x2 ))dµ1 . Condition (67) guarantees that Jϕ∗ = inf{Jϕ∗ (~ µ) : µ ~ ∈ M(σ)} > −∞. 16

In case that µ ~ ∈ M∗ (σ), it is easy to check that Z Jϕ∗ (~ µ) = 2(I(µ1 ) − I(µ1 , µ2 ) + I(µ2 ) +

(69)

ϕdµ1 ) = Jϕ (~ µ).

A vector measure ~λ ∈ M(σ) is said to be extremal if Jϕ∗ (~λ) = Jϕ∗ . The next lemma complements, in the present setting, results from [31]. Lemma 4.1. Let ϕ satisfy (67) and let σ ∈ M+ (R− ), supp(σ) = R− , |σ| > 1, be such that for every compact subset K ⊂ R− we have that P σ|K is continuous on C. The following statements are equivalent and have the same unique solution: (A) There exists ~λ ∈ M(σ) such that Jϕ∗ (~λ) = Jϕ∗ . (B) There exists ~λ ∈ M(σ) such that for all ~ν ∈ M(σ) Z Z Z ~λ ~λ ~ ~ W d(~ν − λ) := W1 d(ν1 − λ1 ) + W2λ d(ν2 − λ2 ) ≥ 0. (C) There exist ~λ = (λ1 , λ2 ) ∈ M(σ) and constants γ1 , γ2 such that (i)  = γ1 , x ∈ supp(λ1 ), ~λ λ1 λ2 W1 (x) = 2V (x) − V (x) + ϕ(x) ≥ γ1 , x ∈ R+ , (ii) ~ W2λ (x)

λ2

λ1



= 2V (x) − V (x)

≤ γ2 , x ∈ supp(λ2 ), ≥ γ2 , x ∈ supp(σ − λ2 ).

The constants γ1 , γ2 are uniquely determined as well. V λ1 and V λ2 are continuous on C. Proof. As shown in [31, Theorem 2.6], the functional Jϕ∗ is lower semicontinuous and strictly convex on M(σ), from which the existence of a unique solution to (A) is guaranteed by [31, Corollary 2.7]. Let us show that problems (A) and (B) are equivalent and have the same solution. Take ~λ, ~ν ∈ M(σ) and ε, 0 ≤ ε ≤ 1. Define ~νε = ε~ν +(1−ε)~λ. Straightforward calculations yield Z ~ 2 ~ ~ Jϕ∗ (~νε ) − Jϕ∗ (λ) = ε J0 (~ν − λ) + 2ε W λ · d(~ν − ~λ), where J0 (~ν − ~λ) is the functional applied to ~ν − ~λ with ϕ∗ ≡ 0. Assume that ~λ is extremal. From the last formula it follows that Z ~ 2 ~ ε J0 (~ν − λ) + 2ε W λ · d(~ν − ~λ) ≥ 0. Dividing by ε and letting ε → 0, we have Z ~ (70) W λ · d(~ν − ~λ) ≥ 0,

~ν ∈ M(σ),

so (A) implies (B). Taking ε = 1, we get Jϕ∗ (~ν ) − Jϕ∗ (~λ) = J0 (~ν − ~λ) + 2

Z

~ W λ · d(~ν − ~λ).

Since the matrix A is positive definite, it follows that J0 (~ν − ~λ) ≥ 0 with equality if and only if ~ν = ~λ. For the proof see [31, Propositions 3.1, 3.5] (see also [16, Theorem 2.5]). Therefore, (B) implies (A). 17

If ~ν ∈ M(σ) is also extremal, then J0 (~ν − ~λ) + 2

Z

~

W λ d(~ν − ~λ) = 0.

Taking (70) into consideration it follows that J0 (~ν − µ ~ ) = 0 and, consequently ~ν = ~λ. Therefore, an extremal vector measure is unique. Now, let us prove that any solution to (C) solves (B). Let ~λ = (λ1 , λ2 )t verify (C) and take ~ν = (ν1 , ν2 )t ∈ M(σ). Notice that Z Z Z ~ ~ ~ W λ d(~ν − ~λ) = W1λ d(ν1 − λ1 ) + W2λ d(ν2 − λ2 ). From (C), condition (i), it follows that Z Z Z ~λ ~λ ~ W1 d(ν1 − λ1 ) = W1 dν1 − W1λ dλ1 ≥ γ1 − γ1 = 0. On the other hand, |λ2 | = |ν2 | = 1; therefore, Z Z ~λ ~ W2 d(ν2 − λ2 ) = (W2λ − γ2 ) d(ν2 − λ2 ). Define ~

~

E+ = {t ∈ supp(σ) : W2λ (t) − γ2 > 0},

E− = {t ∈ supp(σ) : W2λ (t) − γ2 < 0}.

According to (ii) in (C), λ2 (E+ ) = 0, so Z Z ~λ (W2 − γ2 )d(ν2 − λ2 ) ≥ E+

~

(W2λ − γ2 )dν2 ≥ 0.

E+

Additionally, since ν2 ≤ σ and (σ − λ2 )(E− ) = 0, Z Z Z ~λ ~λ (W2 − γ2 )d(ν2 − λ2 ) = (W2 − γ2 )d(ν2 − σ) + E−

E−

~

(W2λ − γ2 )d(σ − λ2 ) ≥ 0.

E−

Putting these relations together, we obtain Z ~ W λ d(~ν − ~λ) ≥ 0,

ν ∈ M(σ),

as claimed. Therefore, (C) has a unique solution. It remains to show that (B) implies (C). First, let us prove that V λ2 is continuous on C. Obviously, V λ2 is continuous on C \ R− . By Fatou’s lemma it readily follows that it is lower semi-continuous on R− . Therefore, it is sufficient to show that it is also upper semicontinuous on this set. Choose x0 ≤ 0. If x0 < 0, take K = [−2x0 , 0]; if x0 = 0, take K = [−1, 0]. Then p Z 1 + y2 λ2 V (x) = log dλ2 (y) |x − y| R− \K p p Z Z 1 + y2 1 + y2 + log dσ(y) − log d(σ − λ2 )(y). |x − y| |x − y| K K The first term in the right hand side is obviously continuous at x0 , which is at a positive distance from R− \ K; the second one is also continuous at x0 , since by hypothesis P σ|K is continuous on R− . Finally, taking into consideration that σ − λ2 is a positive measure, the last term (with the minus sign included) is upper semi-continuous on R− . Therefore, the sum of the three terms is upper semi-continuous at x0 . 18

Set γ1 :=

1 2

Z

~

W1λ dλ1 .

Let us prove that ~

W1λ (x) ≥ γ1

(71)

quasi-everywhere on R+ ,

where “quasi-everywhere” means except on a set of capacity zero. If this was not so, there ~ would exists a compact subset K1 ⊂ R+ , cap(K1 ) > 0, such that W1λ (x) < γ1 , x ∈ K1 . Taking ν1 ∈ M+ 2 (R+ ), supp(ν1 ) ⊂ K1 , and ν2 = λ2 , we obtain Z Z ~ ~ W λ d(~ν − ~λ) = W1λ d(ν1 − λ1 ) < 2γ1 − 2γ1 = 0, which contradicts (B). Now, we prove that ~

W1λ (x) ≤ γ1 ,

x ∈ supp(λ1 ). ~

To the contrary, assume that there exists x0 ∈ supp(λ1 ) such that W1λ (x0 ) > γ1 . By the lower ~ semi-continuity of W1λ on R+ (recall that V λ2 and ϕ are continuous) it follows that there exists ~ δ > 0 such that W1λ (x) > γ1 , |x − x0 | ≤ δ. Take K2 = supp(λ1 ) ∩ {x : |x − x0 | ≤ δ}. Then λ1 (K2 ) > 0 and Z Z ~λ ~ W1 dλ1 + W1λ dλ1 > γ1 (λ1 (supp(λ1 ) \ K2 ) + λ1 (K2 )) = 2γ1 , 2γ1 = supp(λ1 )\K2

K2

which is also a contradiction. From (71), reasoning as in [39, Theorem 5.4.1], it follows that ~ W1λ ≥ γ1 on all R+ . Hence, statement (i) of (C) is obtained. We have also obtained that V1λ1 is continuous on all C since it is continuous on supp(λ1 ). Set ~ γ2 := sup{γ ∈ R : W2λ ≥ γ (σ − λ2 ) a.e.}. Obviously, this supreme is a maximum. Suppose that there exists x0 ∈ supp(λ2 ) such that ~ W2λ (x0 ) > γ > γ2 . By the definition of γ2 , there exists a compact K1 ⊂ supp(σ − λ2 ), ~ ~ such that W2λ (x) < γ, x ∈ K1 , and (σ − λ2 )(K1 ) > 0. On the other hand, W2λ (x) is lower semi-continuous on R− (in fact continuous), so there exists δ > 0 sufficiently small such that ~ W2λ (x) > γ for |x − x0 | < δ, and by the same token there exists a compact set K2 with ~ λ2 (K2 ) > 0, such that W2λ (x) > γ for x ∈ K2 . Obviously, K1 ∩ K2 = ∅. Choose α, β ∈ (0, 1) such that β(σ − λ2 )(K1 ) = αλ2 (K2 ). Define a signed measure η equal to −αλ2 on K2 , equal to β(σ − λ2 ) on K1 , and zero otherwise. It is easy to check that λ2 + η is a positive measure of total mass 1 which is bounded by σ. Define ~ν := (λ1 , λ2 + η)t ∈ M(σ). Then Z Z ~ ~ W λ d(~ν − ~λ) = W2λ dη < γβ(σ − λ)(K1 ) − γαλ2 (K2 ) = 0, ~

in contradiction with (B). From the continuity of W2λ on C, the inequality in the second part of (C) − (ii) holds for all x ∈ supp(σ − λ2 ). Therefore, (C) has been proved. From the uniqueness of ~λ and the fact that supp(σ − λ) ∩ supp(λ2 ) 6= ∅ it readily follows that γ1 , γ2 are uniquely determined.  Lemma 4.2. With the assumptions of Lemma 4.1, let ~λ be extremal. Then, supp(λ2 ) is connected. If xϕ0 (x) is an increasing function on R+ then supp(λ1 ) is also connected. If (72)

lim (ϕ(x) − 4 log x) = +∞,

x→+∞

19

then supp(λ1 ) is a compact set, Z (73) log(1 + y 2 )dλ1 (y) < +∞,

Z

log(1 + y 2 )dλ2 (y) < +∞,

and in (C − ii), Z (74)

γ2 =

1 log(1 + y )dλ2 (y) − 2 2

Z

log(1 + y 2 )dλ1 (y).

Proof. Assume that supp(λ2 ) is not connected. Then, there exist x1 , x2 ∈ supp(λ2 ), x2 < 0, such that the interval (x1 , x2 ) ∩ supp(λ2 ) = ∅. Straightforward calculations lead to   Z Z 0  ydλ2 (y) −ydλ1 (y) d ~λ x W2 (x) =2 + < 0, x ∈ (x1 , x2 ). dx (x − y)2 (x − y)2
0 This implies that 2V λ2 (x) − V λ1 (x) cannot change sign from plus to minus on (x1 , x2 ), which contradicts (C − ii). On the other hand, Z Z  0 dλ2 (y) dλ1 (y) ~λ −x + xϕ0 (x) x W1 (x) = 2x y−x y−x and   Z Z Z Z dλ2 (y) 0 ydλ1 (y) −ydλ2 (y) dλ1 (y) −x =2 + > 0, x ∈ R+ \ supp(λ1 ). 2x 2 y−x y−x (x − y) (x − y)2  0 ~ Therefore, if xϕ0 (x) in increasing on R+ , the function x W1λ (x) is increasing on any subin 0 ~ terval contained in R+ \ supp(λ1 ). Therefore, on any such subinterval W1λ (x) cannot change sign from plus to minus and the connectedness of supp(λ1 ) readily follows using (C − i). Let us show that condition (72) implies that supp(λ1 ) is a compact set. Indeed, according to (C − i), ( p p Z Z 1 + y 2 λ1 1 + y 2 λ2 ϕ(x) = γ1 /4, x ∈ supp(λ1 ), d (y) − log d (y) + log |x − y| 2 |x − y| 4 4 ≥ γ1 /4, x ∈ R+ , and (λ1 , λ2 ) is the only pair of measures in M(σ) satisfying (C). The assumption on ϕ implies that ! p Z 1 + y 2 λ2 ϕ(x) lim − log d (y) + − log x = +∞. x→+∞ |x − y| 4 4 e with compact According to [48, Theorem 1.1.3], there exists a unique probability measure λ support contained in R+ and a unique constant γ e such that p  Z e 1 + y 2 λ2 ϕ(x) = γ e e, x ∈ supp(λ), λ d (y) + P (x) − log ≥γ e, x ∈ R+ . |x − y| 4 4 e has compact support and From uniqueness, it follows that λ1 = 2λ Z log(1 + y 2 )dλ1 (y) = γ1 − 4e γ < +∞. 20

If supp(λ2 ) is compact, the other relation in (73) is immediate. Should supp(λ2 ) be unbounded, since |λ1 | = 2, |λ2 | = 1, from the first part of (C − ii) we have p p Z Z 1 + y2 1 + y2 (75) 2 log dλ2 (y) − log dλ1 (y) ≤ γ2 , x ∈ supp(λ2 ). |1 − (y/x)| |1 − (y/x)| This implies that Z lim sup

2

x→∞,x∈supp(λ2 )

p Z 1 + y2 1 log dλ2 (y) ≤ γ2 + log(1 + y 2 )dλ1 (y) |1 − (y/x)| 2

Using (66), we get p 1 + y2 |x| ≥√ . |1 − (y/x)| 1 + x2 Therefore, for all x ≤ −1 p 1 + y2 log ≥ −(log 2)/2. |1 − (y/x)| Applying Fatou’s lemma, we get p Z Z 1 + y2 2 dλ2 (y) ≤ log(1 + y )dλ2 (y) = 2 lim inf log |1 − (y/x)| x→∞,x∈supp(λ2 ) p Z Z 1 1 + y2 dλ2 (y) ≤ γ2 + log(1 + y 2 )dλ1 (y) < +∞. lim inf 2 log |1 − (y/x)| 2 x→∞,x∈supp(λ2 ) With this we conclude the proof.



Proof of Theorem 2.1. The first part follows directly from Lemma 4.1 and (73). In fact, regarding statements (A) see (69). Then, (B) and (C) follow immediately using the connection between the potentials given by the formulas 2V λ1 − V λ2 + ϕ = 2P λ1 − P λ2 + ϕ + C1 ,

2V λ2 − V λ1 = 2P λ2 − P λ1 + C2 , .

where Z 1 log(1 + y 2 )dλ2 (y), C1 = log(1 + y )dλ1 (y) − 2 Z Z 1 2 C2 = log(1 + y )dλ2 (y) − log(1 + y 2 )dλ1 (y). 2 Z

2

Thus w1 = γ 1 − C 1 ,

w2 = γ2 − C2 .

The complementary statements of the theorem are direct consequence of the remaining assertions of Lemma 4.2. Regarding the value of w2 for the case when supp(λ2 ) ∩ supp(σ − λ2 ) is unbounded notice that in this case there is at least a sequence (xn )n∈Z+ ⊂ supp(λ2 )∩supp(σ−λ2 ), limn xn = −∞ for which equality holds in (75). This implies, taking lim inf along this sequence of points and using Vitali’s theorem (see [44, pp. 134-135]), that Z Z 1 2 γ2 = log(1 + y )dλ2 (y) − log(1 + y 2 )dλ1 (y). 2 We are done.

 21

5. Proof of Theorem 2.2 Proof. Suppose that for some Λ ⊂ N, lim νQn = λ∗1 ,

(76)

lim νQn,2 = λ∗2 .

n∈Λ

n∈Λ

in the weak-* topology. That is Z Z (77) lim f dνQn = f dλ∗1 ,

Z

Z lim

n∈Λ

n∈Λ

f dνQn,2 =

f dλ∗2

for every bounded, continuous, real valued function f on R. Since the zeros of Qn and Qn,2 lie in R+ and R− , respectively, it follows that |λ∗1 | = 1 in R+ ∪ {+∞} and |λ∗2 | = 1 in R− ∪ {−∞}. Once we prove that λ∗1 and λ∗2 verify the second part of (18) we obtain that the limiting measures have no mass at ∞ and if λ∗2 ≤ σ we obtain that (2λ∗1 , λ∗2 ) ∈ M∗ (σ). We also prove that (2λ∗1 , λ∗2 ) ∈ M∗ (σ) solves the extremal problem (C) in Theorem 2.1. Then from unicity it follows that all convergent subsequences have the same limit. This basically settles Theorem 2.2. Between two consecutive mass points of the discrete measure σ2,n there may be at most one zero of Qn,2 . Choose T ∈ (−∞, 0), then from (14) it follows that Z Z Z X 1 dνQn,2 ≤ lim lim sup d δξk,n = dσ. n n [T,0] n [T,0] [T,0] k≥0

This, together with the properties of σ, and the second part of (77) shows that λ∗2 |R− ≤ σ. According to (11), Z Z q Y |Qn (x)|2 |Q(x)|2 0 0 Cn σ (dn x)dx ≤ Cn σ (dn x)dx, Cn = 1 + x2k , |Qn,2 (x)| |Qn,2 (x)| Qn,2 (xk )=0

for any monic polynomials Q, deg Q = 2n. Notice that p Z |Qn,2 (x)| 1 1 + y2 log = − log d νQn,2 (y). n Cn |x − y| Since the kernel is bounded and continuous on R− for all x > 0, from (76) we obtain that p   Z |Qn,2 (x)| 1/2 1 1 ∗ 1 1 + y2 ∗ (78) lim log =− log dλ2 (y) = − V λ2 (x) n∈Λ 2n Cn 4 |x − y| 4 uniformly with respect to x on each compact subset of R+ \ {0}. For x = 0 we also have pointwise convergence since the kernel is uniformly bounded from below and we can use truncated kernels and Lebesgue’s monotone convergence theorem to obtain convergence at that point. In fact, there is uniform convergence on any compact subset of R+ because ∗ λ∗2 |R− ≤ σ and we can prove that V λ2 is continuous on C as we did to prove the same property for V λ2 . (Notice that the kernel takes value zero at −∞ for any x ∈ R+ so the integral in (78) is the same integrating over R− or R− ∪ {−∞}.) Using (15), we also obtain lim n

1 1 log 0 = ϕ(x), n σ1 (dn x)

uniformly on each compact subset of R+ . Consequently,   |Qn,2 (x)| 1/2 1 1 ∗ lim log = (ϕ(x) − V λ2 (x)), n∈Λ 2n Cn σ 0 (dn x) 4 22

uniformly on each compact subset of R+ . From (16) it follows that   ∗ (79) lim ϕ(x) − V λ2 (x) − 4 log x = +∞. x→+∞

The properties shown above allow us to apply the Lemma and Theorem in [27]2. It follows that λ∗1 solves the extremal problem  1 = w1∗ , x ∈ supp(λ∗1 ), λ∗2 λ∗1 (80) P (x) + (ϕ(x) − V (x)) ≥ w1∗ , x ∈ R+ , 4 for some constant w1∗ , and Z (81)

lim

n∈Λ

|Qn (x)|2 Cn σ 0 (dn x)dx |Qn,2 (x)|

1/4n

∗

= e−w1 .

Moreover, the same results guarantee that supp(λ∗1 ) is a compact subset of [0, +∞) (see R comments after [27, Lemma, p.121]). This in turn implies that log(1 + y 2 )dλ∗1 (y) < ∞ and, ∗ in particular, |λ∗1 | = 1 in R+ . The continuity of P λ1 on supp(λ∗1 ) follows from (80), so this potential is also continuous on C. Formula (80) may be easily rewritten as  = γ1∗ , x ∈ supp(2λ∗1 ), λ∗2 2λ∗1 (82) 2V (x) − V (x) + ϕ(x) ≥ γ1∗ , x ∈ R+ , R with γ1∗ = 4w1∗ + 2 log(1 + y 2 )dλ∗1 (y), which is the first part of (C) in Lemma 4.1 for the pair (2λ∗1 , λ∗2 ). The varying discrete measure with respect to which Qn,2 is orthogonal, see (12), is Z ∞ X |Qn (x)|2 Cn σ 0 (dn x)dx βk . ηn,k δξk,n (t), ηn,k = |Qn (ξk,n )| R+ |Qn,2 (x)| x − ξk,n k=1

1/n

Taking into consideration that |x − ξk,n | > |t1 /dn |, that limn dn = 1, the compactness of supp(λ∗1 ), and (81) one has  Z 1/2n |Qn (x)|2 Cn σ 0 (dn x)dx ∗ (83) lim βk = e−2w1 n∈Λ |Qn,2 (x)| x − ξk,n uniformly on k in {k : ξk,n ∈ K} for every compact K ⊂ (−∞, 0]. On the other hand, from ∗ (76) and the continuity of P λ1 it also follows that lim |Qn (x)|1/2n = e−P

(84)

λ∗ 1

(x)

n∈Λ ∗

Set g(x) = 2w1∗ − P λ1 (x). Using (83)–(84) we have (85)

1/n lim η n∈Λ n,k

− e−g(ξn,k ) = 0

uniformly on k in {k : τn,k ∈ K} for every compact K ⊂ (−∞, 0]. Relation (85) together with the assumptions of Theorem 2.2 complete the requirements for the use of [35, Theorem 7.1] (see also [36]) except that in our case limx→−∞ (g(x) − log |x|) = 2w1 6= +∞, while in [35, Theorem 7.1] this limit is required to be +∞. However, the result remains valid with small variations. We cannot assert the supp(λ∗2 ) is a compact set (and in λ∗ 2Actually, in the proof of those results the slightly stronger condition lim x→+∞ (ϕ(x) − V 2 (x))/4 log x > 1

is required; however (79) is also sufficient. 23

∗

general it is not) and instead of using apriori the potential P λ2 to state the conclusion we ∗ must make use of V λ2 . Thus, we deduce that λ∗2 satisfies the equilibrium conditions  ∗ ∗ ≤ w2∗ , x ∈ supp(λ∗2 ), (86) V λ2 (x) − P λ1 (x) + 2w1∗ ≥ w2∗ , x ∈ supp(σ − λ∗2 ), for some constant w2∗ and Z (87)

lim

n∈Λ

Q2n,2 (t) 1 Cn2 |Qn (t)|

Z

Q2n (x)

Cn σ 0 (dn x)dx dσ2,n (t) |Qn,2 (x)| |x − t|

!1/2n ∗

= e−w2 .

(The Cn and Cn2 , which obviously simplify, come from the need of balancing Qn,2 so as to use ∗ ν in the calculations the modified potential V Qn,2 and its limit V λ2 . We wrote the left hand of (87) in that form to underline this fact.) Reasoning as in theR last part of the proof of Lemma 4.2, the first inequality in (86) on supp(λ∗2 ) implies that log(1 + y 2 )dλ∗2 < +∞. Consequently, |λ∗2 | = 1 on R− and (2λ∗1 , λ∗2 ) ∈ M∗ (σ). Moreover, relations (86) can be rewritten as follows  ≤ γ2∗ , x ∈ supp(λ∗2 ), 2λ∗1 λ∗2 (88) 2V (x) − V (x) ≥ γ2∗ , x ∈ supp(σ − λ∗2 ), R with γ2∗ = 2w2∗ − 4w1∗ − log(1 + y 2 )dλ∗1 (y). Since (2λ∗1 , λ∗2 ) fulfills relations (82) and (88) we have that this pair of measures is the unique extremal solution of Lemma 4.1 (or what is the same in this case of Theorem 2.1). This is true for any subsequence of zero counting measures satisfying (76). So the sequences converge to λ1 /2 and λ2 respectively, as stated. Notice that the limits in (81) and (87) also the constants w1∗ , w2∗ on the right R exist because 2 hand are uniquely determined. Using that log(1 + y )dλ2 < +∞, the definition of Cn , and (14), we obtain √ R 2 (89) lim Cn1/n = e log 1+y dλ2 (y) . n

So Z lim n

|Qn (x)|2 0 σ (dn x)dx |Qn,2 (x)|

1/n

From (80) it is easy to check that w1 = 4w1∗ + gously, from (87) and (89) we get

R

∗

= e−(4w1 + log

R

√ log

1+y 2 dλ2 (y))

.

p 1 + y 2 dλ2 (y) and (23) follows. Analo-

!1/n √ R Q2n (x) σ 0 (dn x)dx ∗ 2 lim dσ2,n (t) = e−2w2 + log 1+y dλ2 (y) . n∈Λ |Qn,2 (x)| |x − t| R Using (86), we get w2 = 2w2∗ − 4w1∗ − log(1 + y 2 )dλ2 . Thus, Z p ∗ w1 + w2 = 2w2 − log 1 + y 2 dλ2 (y), Z

Q2n,2 (t) |Qn (t)|

Z

as needed to complete the proof of (24) and the theorem.



Acknowledgements The authors wish to acknowledge useful discussions with B. Beckermann and E. A. Rakhmanov on several aspects of this paper. 24

REFERENCES [1] M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 1972. [2] A. I. Aptekarev, J. Arvesu. Asymptotics for multiple Meixner polynomials. J. Math. Anal. Appl. 411 (2014), 485-505. [3] A. I. Aptekarev, P. M. Bleher, and A. B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source, Part II. Comm. Math. Phys. 259 (2005), 367–389. [4] A. I. Aptekarev, A. Branquinho, and W. Van Assche. Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355 (2003), 3887-3914. [5] A. I. Aptekarev and V. A. Kalyagin. Analytic Properties of two-dimensional continued P-fraction expansions with periodical coefficients and their simultaneous Pad´e-Hermite approximants. in Rational Approximation and Applications in Math. and Phys., Proceedings, Lancut 1985 (J. Gilewitz, M. Pindor and W. Siemaszko eds.), Lecture Notes in Math., vol. 1237, Springer - Verlag, (1987), 145 – 160. [6] A. I. Aptekarev and A. B. J. Kuijlaars. Hermite-Pad´e approximations and multiple orthogonal polynomial ensembles. Uspekhi Mat. Nauk 66: 6(402) (2011), 123190; English translation in Russian Math. Surveys 66: 6 (2011), 1133-1199. [7] A. I. Aptekarev and V. G. Lysov. Systems of Markov functions generated by graphs and the asymptotics of their Hermite-Pad´e approximants. Matem. Sb. 201 2 (2010), 29–78; Sb. Math. 201 2 (2010), 183–234. [8] A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov. Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials. Matem. Sb. 202 2 (2011), 3–56; Sb. Math. 202 2 (2011), 155–206. [9] A. I. Aptekarev and H. Stahl. Asymptotics of Hermite-Pad´e polynomials. In Progress in Approximation Theory ( A. Gonchar, E. B. Saff, eds.), Springer-Verlag, Berlin, 1992, pp. 127–167. [10] J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin, and P. D. Miller. Discrete orthogonal polynomials, volume 164 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2007. [11] B. Beckermann, V. Kalyagin, A. C. Matos, and F. Wielonsky. Equilibrium problems with semidefinite interaction matrices and constrained masses. Constr. Approx. 37 (2013), 101–134. [12] M. Bello Hern´ andez, G. L´ opez Lagomasino, and J. M´ınguez Ceniceros. Fourier-Pad´e approximants for Angelesco systems. Constr. Approx. 26 (2007), 339-359. [13] P. Bleher, S. Delvaux, and A. B. J. Kuijlaars, Random matrix model with external source and a constrained vector equilibrium problem. Comm. Pure Appl. Math. 64 (2011), 116–160. [14] P. M. Bleher and A. B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source I. Comm. Math. Phys. 252 (2004), 43–76. [15] J. Bustamante. Asymptotics for Angelesco Nikishin systems. J. Approx. Theory 85 (1996), 43–68. [16] U. Cegrell, S. Kolodziej, and N. Levenberg. Two problems on potential theory with unbounded sets. Math. Scand. 83 (1998), 265–276. [17] P. A. Deift. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. New York University Courant Institute of Mathematical Sciences, New York, 1999. [18] P. D. Dragnev and E.B. Saff. Constrained energy problems with apptications to orthogonal polynomials of a discrete variable. J. D’Analyse Mathematique 72 (1997), 223–259. [19] K. Driver and H. Stahl. Normality in Nikishin systems. Indag. Math. N.S. 5 (1994), 161-187. [20] K. Driver and H. Stahl. Simultaneous rational approximants to Nikishin systems. I. Acta Sci. Math. (Szeged) 60 (1995), 245–263. [21] K. Driver and H. Stahl. Simultaneous rational approximants to Nikishin systems. II. Acta Sci. Math. (Szeged) 61 (1995), 261–284. [22] E. Coussement and W. Van Assche. Multiple orthogonal polynomials associated with the modified Bessel functions of the first kind. Constr. Approx. 19 (2003), 237–263. [23] E. Coussement and W. Van Assche. Asymptotics of multiple orthogonal polynomials associated with the modified Bessel functions of the first kind. J. Comput. Appl. Math. 153 (2003), 141–149. [24] U. Fidalgo and G. L´ opez Lagomasino. Rate of convergence of generalized Hermite-Pad´e approximants of Nikishin systems. Constr. Approx. 23 (2006), 165-196. [25] U. Fidalgo and G. L´ opez Lagomasino. Nikishin systems are perfect. Constr. Approx. 34 (2011), 297-356. [26] U. Fidalgo, G. L´ opez Lagomasino. Nikishin systems are perfect. Case of unbounded and touching supports. J. of Approx. Theory, 163 (2011), 779-811. [27] A. A. Gonchar and E. A. Rakhmanov. Equilibrium measure and the distribution of zeros of extremal polynomials. Math. USSR Sb. 53 (1986), 119–130. 25

[28] A. A. Gonchar and E. A. Rakhmanov. On the convergence of simultaneous Pad´e approximants for systems of functions of Markov type. Trudy Mat. Inst. Steklov. 157 (1981), 31–48; Proc. Steklov Inst. Math. 157 (1983), 31–50. [29] A. A. Gonchar and E. A. Rakhmanov. On the equilibrium problem for vector potentials. Uspekhi Mat. Nauk 40 (1985), no. 4, 155–156; Russian Math. Surveys 40 (1985), no. 4, 183–184. [30] A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin. Hermite–Pad´e approximants for systems of Markov– type functions. Matem. Sb. 188 (1997), 33–58; Sb. Math. 188 (1997), 33–58. [31] A. Hardy and A. Kuijlaars. Weakly admissible vector equilibrium problems. J. Approx. Theory 164 (2012), 854–868. [32] A. B. J. Kuijlaars. Multiple orthogonal polynomial ensembles. In: Recent Trends in Orthogonal Polynomials and Approximation Theory (IWOPA’08). J. Arvesu, F. Marcell´ an, and A. Mart´ınez Finkelshtein Eds.. Contemporary Mathematics, Vol. 507, Amer. Math. Soc., Providence, R.I. 2010. [33] A. B. J. Kuijlaars, A. Mart´ınez-Finkelshtein, and F. Wielonsky. Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights. Comm. Math. Phys., 286 (1):217–275, 2009. [34] A. B. J. Kuijlaars, A. Mart´ınez-Finkelshtein and F. Wielonsky, Non-intersecting squared Bessel paths: critical time and double scaling limit, Comm. Math. Phys., 308, 227–279, 2011. [35] A. B. J. Kuijlaars and E. A. Rakhmanov. Zero distributions for discrete orthogonal polynomials. J. Comp. Appl. Math. 99 1998, 255–274. [36] A. B. J. Kuijlaars and W. Van Assche. Extremal polynomials on discrete sets. Proc. London Math. Soc. 79 (1999), 191-221. [37] E. M. Nikishin. On simultaneous Pad´e approximants. Matem. Sb. 113 (1980), 499–519 (Russian); English translation in Math. USSR Sb. 41 (1982), 409–425. [38] E. M. Nikishin. Asymptotic behavior of linear forms for simultaneous Pad´e approximants. Izv. Vyssh. Uchebn. Zaved. Mat. (1986), no. 2, 33–41; Soviet Math. 30 (1986), no. 2, 43–52. [39] E. M. Nikishin and V. N. Sorokin. Rational Approximation and Orthogonality. Transl. of Math. Monographs Vol. 92, Amer. Math. Soc., Providence, R.I. 1991. [40] L. Pastur. The spectrum of random matrices. Theoret. Mat. Fiz. 10 (1972), 102–112. [41] E. A. Rakhmanov. Equilibtium measure and the distribution of zeros of exremal polynomials of a discrete variable. Matem. Sb. 187 (1996), 109–124; Sbornik Mathematics 187 (1996), 1213–1228. [42] E. A. Rakhmanov. The asymptotics of Hermite-Pad´e polynomials for two Markov-type functions. Matem. Sb. 202 1 (2011), 127–134; Sb. Math. 202 1 (2011), 127–134. [43] E. A. Rakhmanov and S. P. Suetin. Asymptotic behaviour of the Hermite-Pad´e polynomials of the first kind for a pair of functions forming a Nikishin system. Uspekhi Mat. Nauk 67: 5(407) (2012), 177–178; Russian Math. Surveys, 67: 5 (2012), 954–956. [44] W. Rudin. Real and Complex Analysis. McGraw-Hill Series in Higher Math., N. York, 1966. [45] P. Simeonov. A weighted energy problem for a class of admissible weights. Houston J. Math., 31 (2005), pp. 1245-1260 [46] V. N. Sorokin. Generalized Pollaczek polynomials. Mat. Sb., 200 (2009), 113–130. [47] V. N. Sorokin. On multiple orthogonal polynomials for discrete Meixner measures. Matem. Sb. 201 (2010), 137–160; Sb. Math. 201 (2010), 1539–1561. [48] E. B. Saff and V. Totik. Logartihmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997. [49] H. Stahl. Simultaneous rational approximants. In: Proceedings of Computaional Mathematics and Function Theory (CMFT’94), World Scientific Publishing Co. R.M Ali, St. Rusheweyh, and E.B. Saff Eds., 1995. [50] H. Stahl and V. Totik, General Orthogonal Polynomials. Enc. Math. Vol. 43, Cambridge University Press, Cambridge, 1992. [51] W. Van Assche, J. S. Geronimo, and A. B. J. Kuijlaars. Riemann-Hilbert problems for multiple orthogonal polynomials. In Special functions 2000: current perspective and future directions (Tempe, AZ), volume 30 of NATO Sci. Ser. II Math. Phys. Chem., pages 23–59. Kluwer Acad. Publ., Dordrecht, 2001.

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(Aptekarev) Keldysh Institute of Applied Mathematics, Moscow, Russia E-mail address, Aptekarev: [email protected] ´s, (L´ opez-Lagomasino) Department of Mathematics, Universidad Carlos III de Madrid, Legane Spain E-mail address, L´ opez: [email protected] (Mart´ınez-Finkelshtein) Department of Mathematics, University of Almer´ıa, Almer´ıa, Spain E-mail address, Mart´ınez: [email protected]

27