Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials
A recent result of S.-Y. Lee and M. Yang state that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2023 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211864 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials. Sergey Berezin, Arno B.J. Kuijlaars and Iván Parra. SIGMA 19 (2023), 020, 18 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860254227709820928 |
|---|---|
| author | Berezin, Sergey Kuijlaars, Arno B.J. Parra, Iván |
| author_facet | Berezin, Sergey Kuijlaars, Arno B.J. Parra, Iván |
| citation_txt | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials. Sergey Berezin, Arno B.J. Kuijlaars and Iván Parra. SIGMA 19 (2023), 020, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A recent result of S.-Y. Lee and M. Yang state that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integers. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique.
|
| first_indexed | 2026-03-21T07:04:37Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 020, 18 pages
Planar Orthogonal Polynomials
as Type I Multiple Orthogonal Polynomials
Sergey BEREZIN ab, Arno B.J. KUIJLAARS a and Iván PARRA a
a) Department of Mathematics, Katholieke Universiteit Leuven,
Celestijnenlaan 200B box 2400, 3001 Leuven, Belgium
E-mail: sergey.berezin@kuleuven.be, arno.kuijlaars@kuleuven.be, ivan.parra@kuleuven.be
b) St. Petersburg Department of V.A. Steklov Mathematical Institute of RAS,
Fontanka 27, 191023 St. Petersburg, Russia
E-mail: serberezin@math.huji.ac.il
Received December 14, 2022, in final form March 21, 2023; Published online April 12, 2023
https://doi.org/10.3842/SIGMA.2023.020
Abstract. A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal
polynomials orthogonal with respect to a modified Gaussian measure are multiple orthog-
onal polynomials of type II on a contour in the complex plane. We show that the same
polynomials are also type I orthogonal polynomials on a contour, provided the exponents
in the weight are integer. From this orthogonality, we derive several equivalent Riemann–
Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which
we establish using a new technique.
Key words: planar orthogonal polynomials; multiple orthogonal polynomials; Riemann–
Hilbert problems; Hermite–Padé approximation; normal matrix model
2020 Mathematics Subject Classification: 42C05; 30E25; 41A21
Dedicated to Alexander Its
on the occasion of his 70th birthday
1 Introduction
This work is inspired by Lee and Yang’s paper [17], which showed that planar orthogonal poly-
nomials can be viewed as multiple orthogonal polynomials of type II on a contour in the complex
plane. Their work extends an earlier result of Balogh, Bertola, Lee, and McLaughlin [3].
We show that the same polynomials are also multiple orthogonal polynomials of type I if
the exponents in the weight are positive integers, unlike in the situation studied in [17], where
these exponents are arbitrary positive real numbers. We also present a novel, more transparent,
technique to transform planar orthogonality into orthogonality on a contour. Before we begin,
note that the title of our paper differs from the title of [17] only in one letter, yet this makes
a considerable difference in the arguments used.
The polynomials in question are orthogonal with respect to a modified Gaussian measure,
µW (dz) =
1
π
|W (z)|2e−|z|2 Leb(dz), (1.1)
This paper is a contribution to the Special Issue on Evolution Equations, Exactly Solvable Mod-
els and Random Matrices in honor of Alexander Its’ 70th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Its.html
mailto:sergey.berezin@kuleuven.be
mailto:arno.kuijlaars@kuleuven.be
mailto:ivan.parra@kuleuven.be
mailto:serberezin@math.huji.ac.il
https://doi.org/10.3842/SIGMA.2023.020
https://www.emis.de/journals/SIGMA/Its.html
2 S. Berezin, A.B.J. Kuijlaars and I. Parra
where Leb denotes the Lebesgue measure on C (identified with R2) and the weight W reads
W (z) =
p∏
j=1
(z − aj)
cj , z ∈ C, (1.2)
where p ∈ N, the cj are positive real numbers, and the aj are distinct complex numbers.
If the cj are not necessarily integer, one needs to specify the branch cuts and fix the branches
in order to render (1.2) unambiguous. This complicates the analysis, and we return to such
a general scenario only episodically. In contrast, if all cj are positive integers, W becomes
a polynomial of degree c =
∑p
j=1 cj and (1.2) extends to the whole complex plane C. This is
the situation of our primary concern.
Denote the scalar product corresponding to (1.1) by
⟨f, g⟩W =
∫
C
f(z)g(z)µW (dz). (1.3)
Then, the n-th degree monic orthogonal polynomial Pn with respect to µW can be uniquely
recovered by solving a linear system of equations for its coefficients,∫
C
Pn(z)z
k µW (dz) = 0, k = 0, . . . , n− 1. (1.4)
The motivation for studying planar orthogonal polynomials comes from the theory of non-
Hermitian random matrices, in particular from those related to the normal matrix model. In this
model, the eigenvalues of an n× n normal matrix have the joint density
1
Zn
∏
j<k
|zk − zj |2
n∏
j=1
e−V (zj), (1.5)
where V is the potential of the model and Zn is a normalization constant. The eigenvalues
form a determinantal point process with the correlation kernel constructed in terms of the
planar orthogonal polynomials orthogonal with respect to the one-particle weight e−V (z). The
case (1.1)–(1.2) corresponds to
V (z) = |z|2 − 2
p∑
j=1
cj log |z − aj |. (1.6)
In particular, for integer cj ’s the probability law corresponding to (1.5)–(1.6) can be interpreted
as that of a Ginibre ensemble of size n+ c conditioned on having an eigenvalue of multiplicity cj
at aj for each j = 1, . . . , p.
The determinantal structure in (1.5) allows for a complete description of the eigenvalue
correlation functions at the finite size n in terms of the correlation kernel, which in turn can
be used to study the large n behavior of both the polynomials and the eigenvalues. In such
studies, one typically replaces V by nV in (1.5) to obtain a balance between the “repulsion” and
“confinement” present in the determinantal model (1.5). We refer to the surveys [7, Section 5],
[13, Chapter 6] and references therein for more information on the normal matrix model.
In the analogous situation of Hermitian random matrices, the eigenvalue correlations are
described by orthogonal polynomials on the real line. The theory of such polynomials is well-
developed, and as a result the corresponding ensembles are understood much better than their
non-Hermitian counterparts. One basic result is that the eigenvalues of Hermitian matrices
and the zeros of the corresponding orthogonal polynomials (both real) have the same limiting
behavior as n → ∞ (e.g., see [9]). More subtle results on the universality of local eigenvalue
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 3
statistics were established using the characterization of orthogonal polynomials on the real line
via a 2× 2 matrix-valued Riemann–Hilbert problem (e.g., see [12]), followed by the Deift–Zhou
steepest descent analysis (e.g., see [10, 11]).
The planar case, on the other hand, is more intricate. One has to distinguish between the
asymptotic behavior of the random eigenvalues governed by (1.5) (with V replaced by nV ) and
the limiting behavior of the zeros of the corresponding orthogonal polynomials. While it is known
that the eigenvalues fill out a two-dimensional domain called the droplet, the understanding of
the asymptotic behavior of the zeros of the planar orthogonal polynomials is rather limited.
Results exist when the classical Hermite, Laguerre and Gegenbauer polynomials appear as the
planar orthogonal polynomials (e.g., see [1, 15, 21]), as well as for some special cases where
the planar orthogonality can be reformulated as (multiple) orthogonality on a contour and the
Riemann–Hilbert techniques can be used (e.g., see [3, 4, 5, 6, 8, 16]). For example, in [3] the
situation (1.1)–(1.2) with p = 1 is considered. In this case, the Hermitian planar orthogonality
can be transformed to non-Hermitian orthogonality on a contour due to a special identity [3,
Lemma 3.1], and rigorous analysis is possible. The same identity was used in [22] in a study on
moments of complex Ginibre matrices.
Multiple orthogonality plays a role in [6, 16, 18], where it can be treated by using large size
Riemann–Hilbert problems (e.g., see [20]).
The common feature of the examples above is that the zeros accumulate along a one-
dimensional curve (or a system of such curves) in the complex plane, known as the mother-
body. We conjecture that this is a general phenomenon for all real analytic potentials, including
those in (1.6). The results of [17, 18] support this conjecture. Indeed, the planar orthogonality
corresponding to (1.5) with V given by
V (z) = n|z|2 − 2
p∑
j=1
cj log |z − aj |
is studied in [17, 18] for the case of fixed cj ’s independent of n. The droplet turns out to be the
unit disk, and the motherbody is supported on a multiple Szegő curve that depends on the aj .
In the scenario when the cj grow linearly with n and p ≥ 2, the Riemann–Hilbert problem
in [18] has not been analyzed successfully yet. The multiple orthogonality of type I that we
discovered, as we will show, leads to several different Riemann–Hilbert problems. Our hope is
that one of them will help to carry through with the steepest descent analysis.
If all cj are integer valued, the planar orthogonal polynomials can also be expressed as ratios
of determinants as shown in [2]. The determinants are growing in size as the cj increase, and
therefore this determinantal formula may not be particularly useful for asymptotic analysis.
We finally remark that the important work of Hedenmalm and Wennman [14] provides the
asymptotic behavior of the planar orthogonal polynomials in the exterior of the droplet and on its
boundary (even slightly inside, under certain assumptions including those of the real analyticity
of the boundary). This, however, does not give any information about the motherbody since it
is inside the droplet (see [14, Remark 1.6 (c)]).
2 Statement of result
Our main result is Theorem 2.1 below. It gives a number of properties that are equivalent to
the planar orthogonality corresponding to (1.1)–(1.4) in the case the cj are positive integers.
Below, we use the conjugate W ∗ of W defined by
W ∗(z) = W (z̄) =
p∏
j=1
(z − aj)
cj , z ∈ C. (2.1)
4 S. Berezin, A.B.J. Kuijlaars and I. Parra
We also use Dz to denote the derivative operator with respect to z, and then W ∗(Dz) is the
differential operator
W ∗(Dz) =
p∏
j=1
(Dz − aj)
cj . (2.2)
Theorem 2.1. Let W be given by (1.2) where all cj are positive integers
(
so that W is a poly-
nomial of degree c =
∑p
j=1 cj
)
, and let the aj be distinct complex numbers. Then, the following
properties are equivalent for a monic polynomial Pn of degree n,
(a) Pn is the planar orthogonal polynomial on C with weight (1.1).
(b) Pn satisfies
1
2πi
∮
γ
Pn(z)W (z)ϕk(z) dz = 0, k = 0, 1, . . . , n− 1, (2.3)
where γ is a closed contour around the origin and
ϕk(z) =
∫ z̄×∞
0
W ∗(u)uke−uz du, (2.4)
where the path of integration in (2.4) goes from 0 to ∞ along the ray arg u = arg z̄.
(c) One has
W ∗(Dz) [Pn(z)W (z)] = O(zn) (2.5)
as z → 0.
(d) There exist polynomials Qj of degQj ≤ cj − 1, for j = 1, . . . , p, such that
Pn(z)W (z) +
p∑
j=1
Qj(z)e
ajz = O
(
zn+c
)
(2.6)
as z → 0.
Remark 2.2. The property (2.3) of the planar orthogonal polynomials has already been ob-
tained by Lee and Yang [17]. They assume that the points aj are distinct, non-zero, and with
different arguments modulo 2π, and show that (2.3) leads to multiple orthogonality of type II
(see also Section 6 below) for real positive cj ’s. In this general situation, because of the branch
cuts, the contour γ in (2.3) has to pass through all aj and can no longer be an arbitrary contour
around the origin.
The proof of Theorem 2.1 relies on the fundamental identity of Lee and Yang [17, Proposi-
tion 1]. We state and prove it in the next section. Our proof is based on a new technique and is
of independent interest. The proof in [17] (see also the proof of [3, Lemma 3.1]) goes as follows.
One first restricts the integral in (1.4) to a large disk DR and then applies Stokes’ theorem to
rewrite the new integral over DR as an integral over the boundary ∂DR. Then, (2.3) follows by
a contour deformation argument and by passing to the limit R → ∞.
In our proof, we only rely on most basic and elementary facts of complex analysis and avoid
the use of Stokes’ theorem. We write the integral over C in polar coordinates and, by analyticity,
deform the angular integral to an integral over γ. The final step is to switch the angular and
the radial integrals by Fubini’s theorem.
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 5
Theorem 2.1 is proved in Section 4. Part (c) of Theorem 2.1 is a very concise representation
of the planar orthogonality in part (a), and thus is of interest in its own right. The equivalence
with part (d) leads directly to the multiple orthogonality of type I as we explain in Section 5.
We show that this orthogonality leads to three different, though closely related, Riemann–
Hilbert problems. The latter uniquely characterize the planar orthogonal polynomial Pn and
the auxiliary polynomials Qj . We point out in passing that none of these Riemann–Hilbert
problems turns out to be related to the Riemann–Hilbert problem in [17] (corresponding to
type II orthogonality) in a canonical way (see also Section 7). In Section 6, we focus on the
type II multiple orthogonality of Lee and Yang [17]. We give the corresponding proof in the
case of polynomial W , which is essentially the same as the proof in [17] but more transparent
since no branch cuts for W are necessary.
3 Fundamental identity
Akin to (2.1), we define the conjugate of a function Q by
Q∗(z) = Q(z̄), z ∈ C. (3.1)
If Q is analytic in a certain domain Ω, then Q∗ is also analytic however in the conjugate
domain Ω∗ = {z ∈ C | z ∈ Ω}. If Q is a polynomial, then Q∗ is the polynomial whose
coefficients are the complex conjugates of those of the polynomial Q.
The following result is due to Lee and Yang [17, Proposition 1]. As already stated above, we
give a different proof. For the sake of clarity, we first deal with the case of polynomial W .
Proposition 3.1. Let P and Q be polynomials and suppose that the cj in (1.2) are positive
integers. Then, the fundamental identity holds,
⟨P,Q⟩W =
1
π
∫
C
P (z)Q(z)|W (z)|2e−|z|2 Leb(dz)
=
1
2πi
∮
γ
P (z)W (z)
∫ z̄×∞
0
W ∗(u)Q∗(u)e−uz dudz, (3.2)
where γ is a simple closed contour that goes around the origin once in the counterclockwise
direction and the path for the u integral goes from 0 to ∞ along the ray arg u = arg z̄.
Proof. Write the left-hand side of (3.2) in polar coordinates,
⟨P,Q⟩W =
1
πi
∫ ∞
0
∮
Cr
P (z)Q∗(z)|W (z)|2dz
z
re−r2dr, (3.3)
where the z-integral is taken along the circle Cr of radius r around the origin. Observe that z =
r2/z for z ∈ Cr. In view of (3.1), we can write the following chain of identities,
|W (z)|2 = W (z)W ∗(z) = W (z)W ∗(r2/z), z ∈ Cr.
Since also Q∗(z) = Q∗(r2/z) for z ∈ Cr, the formula (3.3) becomes
⟨P,Q⟩W =
1
πi
∫ ∞
0
∮
Cr
P (z)Q∗(r2/z)W (z)W ∗(r2/z)dz
z
re−r2dr. (3.4)
Recall that all cj are positive integers, thus W and W ∗ are polynomials and the integrand
in (3.4) is meromorphic in z with a sole pole at z = 0. Then, by Cauchy’s theorem, the contour
6 S. Berezin, A.B.J. Kuijlaars and I. Parra
can be deformed from Cr to a contour γ that goes around the origin in the counterclockwise
direction once and is independent of r. We use Fubini’s theorem to get
⟨P,Q⟩W =
1
πi
∮
γ
P (z)W (z)
∫ ∞
0
Q∗(r2/z)W ∗(r2/z)
re−r2
z
dr dz. (3.5)
Changing variables in the inner integral, u = r2/z, we arrive at (3.2). ■
Our method of proof easily extends to a more general setting. Assume that the cj are
positive real numbers, not necessarily integer as before. As in [17], we restrict ourselves to the
case that all aj are non-zero, distinct, and have different arguments modulo 2π. For convenience,
order the aj so that 0 ≤ arg a1 < arg a2 < · · · < arg ap < 2π. We can still define W by the
same formula (1.2) as earlier, however it is imperative one restrict the domain by making cuts.
Following [17], we choose to cut along the rays
B =
p⋃
j=1
{z ∈ C | z = ajt, t ≥ 1}. (3.6)
The domain C \B is simply connected. Fixing a branch for the power functions in (1.2) ren-
ders W analytic in this domain. Note that such a choice of the branches does not affect |W (z)|2,
which is assumed to be extended by continuity to the whole complex plane C.
To formulate the analogue of Proposition 3.1, we will set
Ω = C \
p⋃
j=1
{z ∈ C | z = ajt, t ≥ 0}, (3.7)
which is a union of sectors separated by the rays arg z = arg aj for j = 1, . . . , p.
a1
a2
a3
a4
γ
Figure 1. The domain Ω in (3.7) and an example contour γ.
The following is [17, Proposition 1].
Proposition 3.2. Suppose that the cj are positive real numbers, but not necessarily integers,
and the aj are non-zero, distinct complex numbers that have different arguments modulo 2π.
Define Ω as in (3.7). Then (3.2) still holds, provided that γ is a counterclockwise-oriented con-
tour in Ω ∪ {a1, . . . , ap}, going around the origin once (e.g., see Figure 1).
Remark 3.3. Note that there is no need to make a cut for a certain aj if the correspond-
ing cj ∈ N. Nevertheless, we do so for the sake of notational convenience.
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 7
Remark 3.4. If a1 = 0, the proposition still holds with the same Ω as in (3.7); however,
one needs to choose γ in Ω ∪ {a2, . . . , ap} instead. Indeed, first note that Remark 3.3 is still
applicable. Hence, without loss of generality c1 /∈ N. One needs to modify B in (3.6) by including
an additional cut from 0 to ∞, transversal to all the other cuts. Then, in a similar way as before,
one can fix a branch of W (z) in the new simply-connected domain C\B. Note that the definition
of Ω in (3.7) does not change, however one of the rays collapses to the point z = 0. Finally,
observe that zc1 effectively cancels from (3.8), and the proof goes through in the same way
except the fact that Q∗(r2/z) still may have a pole at zero and thus γ must not pass through it.
The latter explains why a1 = 0 was excluded from Ω ∪ {a1, . . . , ap}.
Proof of Proposition 3.2. If not all of the cj are integers, then we need branch cuts to de-
fine W and W ∗. For every r > 0, we have
W (z)W ∗(r2/z) = p∏
j=1
(z − aj)
cj
( r2
|aj |2aj − z
z/aj
)cj
, (3.8)
which is initially only analytic in Ω. If r ̸= |aj | for every j = 1, . . . , p, then there is an analytic
continuation across the open straight line segment from aj to
r2
|aj |2aj along the ray arg z = arg aj
for every j = 1, . . . , p. That is, W (z)W ∗(r2/z) has an analytic continuation to
Ωr := Ω ∪
{
(1− t)aj + t
r2
|aj |2
aj
∣∣∣∣ 0 < t < 1
}
.
Observe that r2
|aj |2aj is the image of aj under the reflection about the circle Cr. Thus, Cr is
contained in Ωr.
With these preparations in mind, we follow the proof of Proposition 3.1. The identity (3.4)
still holds. Then, by the above and Cauchy’s theorem, for each r > 0 we are allowed to deform Cr
to a contour γ as in the statement. Since γ is independent of r, we can apply Fubini’s theorem
to (3.4). This yields (3.5), and we can proceed as in the proof of Proposition 3.1. ■
Note that it is not difficult to extend the proofs of Propositions 3.1 and 3.2 to other types of
weights, mutatis-mutandis. For instance, if
µ(dz) =
1
π
|W (z)|2 Leb(dz)(
1 + |z|2
)α , α > 1 + c,
(which generalizes spherical ensembles, e.g., see [7, Section 2.5]), one gets
⟨P,Q⟩W =
1
2πi
∮
γ
P (z)W (z)
∫ z̄×∞
0
Q∗(u)W ∗(u)
(1 + uz)α
dudz,
which holds as long as degP + degQ < 2α− 2c− 2 so that the integral over C converges.
4 Proof of Theorem 2.1
Proof. (a) ⇔ (b): Taking Q(z) = zk in the identity (3.2) and recalling the definition (2.4)
of ϕk, we obtain〈
P, zk
〉
W
=
1
2πi
∮
γ
P (z)W (z)ϕk(z) dz. (4.1)
This identity (4.1) shows that (a) and (b) in Theorem 2.1 are equivalent.
8 S. Berezin, A.B.J. Kuijlaars and I. Parra
(b) ⇔ (c): From basic properties of the Laplace transform, we get∫ z̄×∞
0
W ∗(u)uke−uz du = W ∗(−Dz)(−Dz)
k
[
1
z
]
= W ∗(−Dz)
[
k!
zk+1
]
,
Thus, recalling (2.4), we can write
1
2πi
∮
γ
Pn(z)W (z)ϕk(z) dz =
k!
2πi
∮
γ
Pn(z)W (z)W ∗(−Dz)
[
1
zk+1
]
dz
for any polynomial Pn. Integrating by parts, we find
1
2πi
∮
γ
Pn(z)W (z)ϕk(z)dz =
k!
2πi
∮
γ
W ∗(Dz)[Pn(z)W (z)]
zk+1
dz,
as there are no boundary terms on the closed contour γ and W ∗(Dz) is the adjoint of W
∗(−Dz).
Thus, Pn satisfies (2.3) if and only if
1
2πi
∮
γ
W ∗(Dz)[Pn(z)W (z)]
zk+1
dz = 0, k = 0, . . . , n− 1. (4.2)
The identities (4.2) mean that the coefficient before zk of the polynomial W ∗(Dz)[Pn(z)W (z)]
vanishes for k = 0, . . . , n− 1, and we conclude that items (b) and (c) are equivalent.
In the proof that (c) and (d) are equivalent, we are going to use three basic properties of the
operator W ∗(Dz) from (2.2). These properties are stated separately for the ease of reference.
Property W1. W ∗(Dz) is a linear differential operator with kernel
kerW ∗(Dz) =
{
p∑
j=1
Qj(z)e
ajz
∣∣∣∣ degQj ≤ cj − 1 for j = 1, . . . , p
}
. (4.3)
The kernel is a vector space over C and its dimension is c =
∑p
j=1 cj .
Property W2. If P is a polynomial then W ∗(Dz)[P (z)] is a polynomial and
degW ∗(Dz)[P (z)] = degP (z), (4.4)
provided all aj ̸= 0. If a1 = 0 and degP (z) ≥ c1, then
degW ∗(Dz)[P (z)] = degP (z)− c1. (4.5)
Property W3. W
∗(Dz) reduces the order of vanishing at z = 0 by c = degW , provided the order
of vanishing is greater than or equal to c. Namely, for every non-negative integer n, and analytic
function F with F (z) = O(zn+c) as z → 0, we have
W ∗(Dz)[F (z)] = O(zn) (4.6)
as z → 0.
(d) ⇒ (c): Suppose Pn satisfies (2.6). Then, by applying W ∗(Dz) and using Properties W1
and W3, we obtain
W ∗(Dz)[Pn(z)W (z)] = W ∗(Dz)
[
O
(
zn+c
)]
= O(zn), (4.7)
which is (2.5). Thus, (d) implies (c).
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 9
(c) ⇒ (d): In the proof, we assume without loss of generality that a1 = 0 and c1 ≥ 0. This
includes the case that W has no zero at the origin since one can always set c1 = 0.
Consider the linear mapping π : ker(W ∗(Dz)) → Cc given by
ker(W ∗(Dz)) ∋ Q(z) =
∞∑
j=0
qjz
j π7−→ (q0, . . . , qc1−1, qn+c1 , . . . , qn+c−1) ∈ Cc. (4.8)
We claim that π is injective, which will imply that π is surjective since it is a linear map-
ping between vector spaces of the same dimension c, as follows from (4.8) and Property W1.
Suppose Q ∈ ker(W ∗(Dz)) with π(Q) = (0, . . . , 0) ∈ Cc. Then,
Q(z) =
n+c1−1∑
j=c1
qjz
j +O
(
zn+c
)
(4.9)
as z → 0. From (4.6) and W ∗(Dz)[Q(z)] = 0, it then follows that
W ∗(Dz)
[
n+c1−1∑
j=c1
qjz
j
]
= O(zn) (4.10)
as z → 0. Because of (4.4)–(4.5), we have that the left-hand side of (4.10) is a polynomial of
degree ≤ n − 1, and due to (4.10) it has a zero at z = 0 of order at least n. Hence, (4.10)
vanishes identically, and thus
∑n+c1−1
j=c1
qjz
j belongs to the kernel of W ∗(Dz). By Property W1,
the kernel contains polynomials up to degree c1−1 but no polynomials of higher degrees. Hence,
qj = 0 for j = c1, . . . , n+ c1− 1. Consequently, by (4.9) we have that Q(z) = O(zn+c) as z → 0.
In particular, Q(0) = Q′(0) = · · · = Q(c−1)(0) = 0. Thus Q ∈ ker(W ∗(Dz)) is a solution of
a homogeneous constant coefficient linear ODE of order c with c vanishing initial conditions.
The uniqueness theorem for such ODEs then yields Q ≡ 0, which justifies the claim.
Now, assume that Pn satisfies (2.5), and write
Pn(z)W (z) =
n+c∑
j=0
pjz
j .
Since π is surjective, there is Q ∈ ker(W ∗(Dz)),
Q(z) =
∞∑
j=0
qjz
j ,
such that π(Q) = (−p0, . . . ,−pc1−1,−pn+c1 , . . . ,−pn+c−1). Then,
Pn(z)W (z) +Q(z) =
n+c1−1∑
j=c1
(pj + qj)z
j +O
(
zn+c
)
. (4.11)
Applying W ∗(Dz) to (4.11) and using (2.5), W ∗(Dz)[Q(z)] = 0, and (4.6), we obtain
W ∗(Dz)
[
n+c1−1∑
j=c1
(pj + qj)z
j
]
= O(zn) (4.12)
as z → 0. We find ourselves in the situation similar to that while proving the injectivity of π.
It followed from (4.10) that qj = 0 for j = c1, . . . , n + c1 − 1. In the exactly same way, now it
follows from (4.12) that pj + qj = 0 for j = c1, . . . , n+ c1 − 1.
10 S. Berezin, A.B.J. Kuijlaars and I. Parra
Due to (4.11), we obtain
Pn(z)W (z) +Q(z) = O
(
zn+c
)
,
which is exactly (2.6) because Q ∈ ker(W ∗(Dz)) must be of the form given in (4.3). This shows
that (c) implies (d) and completes the proof of Theorem 2.1. ■
We present an alternative proof of (a) ⇒ (c) by making use of the Fourier transform of the
measure µW in (1.1),
µ̂W (ζ, ζ) =
∫
C
eζz+ζz̄µW (dz). (4.13)
Alternative proof. (a) ⇒ (c): Apply the Wirtinger derivatives
Dζ =
1
2
(
∂
∂u
− i
∂
∂v
)
, Dζ =
1
2
(
∂
∂u
+ i
∂
∂v
)
,
where ζ = u+ iv, to (4.13), and observe that
D
j
ζ
Dk
ζ
[
µ̂W (ζ, ζ)
]
=
∫
C
zjzkeζz+ζzµW (dz). (4.14)
Due to linearity, (4.14) leads to
Pn(Dζ)D
k
ζ
[
µ̂W (ζ, ζ)
]
=
∫
C
Pn(z)z
keζz+ζzµW (dz).
Assuming that Pn is the planar orthogonal polynomial satisfiying (1.4), we have
Pn(Dζ)D
k
ζ
[
µ̂W (ζ, ζ)
]∣∣
ζ=0,ζ=0
= 0, k = 0, . . . , n− 1, (4.15)
where we view ζ and ζ as two independent variables. The latter is permissible as long as the
Fourier transform µ̂W (·, ·), as a function of two complex variables, extends analytically to the
neighborhood of
{
(ξ1, ξ2) ∈ C× C | ξ2 = ξ1
}
. This is the case we encounter below.
For µW from (1.1) with polynomial W , the Fourier transform can be computed as follows,
µ̂W (ζ, ζ̄) =
1
π
∫
C
eζz+ζz|W (z)|2e−|z|2 Leb (dz)
=
1
π
W (Dζ)W
∗(Dζ)
∫
R2
e2(ux+vy)e−x2−y2 dx dy
= W (Dζ)W
∗(Dζ)
[
eζζ
]
, (4.16)
where ζ = u+ iv, z = x+ iy.
Inserting (4.16) into (4.15), and changing the order of the differential operators, we obtain
Dk
ζW
∗(Dζ)(PnW )(Dζ)
[
eζζ
]∣∣
ζ=0,ζ̄=0
= 0, k = 0, . . . , n− 1. (4.17)
Since
(PnW )(Dζ)
[
eζζ
]∣∣
ζ=0
= Pn(ζ)W (ζ),
we see that (4.17) implies
Dk
ζW
∗(Dζ)[Pn(ζ)W (ζ)]
∣∣
ζ=0
= 0, k = 0, . . . , n− 1,
which is equivalent to (2.5). ■
This alternative proof will work for other types of measures µ as long as the Fourier trans-
form µ̂ has a simple enough representation. On the other hand, non-polynomial weights W will
require the use of fractional derivatives, which are known to be non-local operators, causing
substantial complications in this proof.
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 11
5 Type I multiple orthogonality
The property (2.6) in part (d) of Theorem 2.1 can be regarded as a Hermite–Padé approximation
problem of type I at the origin. The general form of such approximation problems is the follow-
ing. Given a collection of analytic functions (or, more generally, formal power series) f0, . . . , fp
at z = 0 and a multi-index n⃗ = (n0, . . . , np) ∈ Np+1, find polynomials Q0, Q1, . . . , Qp of de-
grees degQj ≤ nj − 1, j = 0, 1, . . . , p, such that
p∑
j=0
Qj(z)fj(z) = O
(
z|n⃗|−1
)
as z → 0, where |n⃗| =
∑p
j=0 nj . For more information on Hermite–Padé approximation and
multiple orthogonal polynomials, we refer to [19] and references therein.
The problem (2.6) becomes a Hermite–Padé type I approximation problem for polynomi-
als Q0 = Pn, Q1, . . . , Qp, corresponding to the weights f0(s) = W (s), f1(s) = ea1s, . . . , fp(s) =
eaps and to the multi-index n⃗ = (n+ 1, c1, . . . , cp). This leads directly to the type I multiple or-
thogonality. Indeed, the function on the left-hand side of (2.6) has vanishing Taylor coefficients
up to and including order n+ c− 1. By the Cauchy integral formula, this tells us that (2.6) is
equivalent to
1
2πi
∮
γ
(
Pn(s)
W (s)
sn+c
+
p∑
j=1
Qj(s)
eajs
sn+c
)
sk ds = 0, k = 0, . . . , n+ c− 1, (5.1)
where γ is a simple closed contour around the origin. This is multiple orthogonality of type I
on the contour γ with p + 1 weight functions W (s)/sn+c, ea1s/sn+c, . . . , eaps/sn+c which are
meromorphic with a sole pole at s = 0.
Both multiple orthogonality of type I and type II are characterized by Riemann–Hilbert
problems that were identified by Van Assche, Geronimo, and Kuijlaars in [20]. For the particular
case (5.1), the Riemann–Hilbert problem is of size (p+2)×(p+2) and its jump is on the contour γ.
Below, we will use the following notation. For any oriented contour γ and a function Y defined
in C \ γ, we write Y+ and Y− to denote the limiting values of Y on γ from the left and right
sides, respectively.
Riemann–Hilbert problem (type I multiple orthogonality for Pn, Q1, . . . , Qp). Find solution
Y : C \ γ → C(p+2)×(p+2) such that
(1) Y (z) is analytic for z ∈ C \ γ;
(2) Y+(z) = Y−(z)JY (z) for z ∈ γ, where JY (z) =
1 0 · · · 0 W (z)/zn+c
0 1 · · · 0 ea1z/zn+c
...
...
. . .
...
...
0 0 · · · 1 eapz/zn+c
0 0 · · · 0 1
;
(3) Y (z) =
(
I +O
(
1
z
))
zn 0 · · · 0 0
0 zc1 · · · 0 0
...
...
. . .
...
...
0 0 · · · zcp 0
0 0 · · · 0 z−(n+c)
as z → ∞.
12 S. Berezin, A.B.J. Kuijlaars and I. Parra
The unique solution of the Riemann–Hilbert problem has the polynomials Pn, Q1, . . . , Qp in
its first row. Namely,
Y (z) =
Pn(z) Q1(z) · · · Qp(z)
1
2πi
∮
γ
(
Pn(s)
W (s)
sn+c
+
p∑
j=1
Qj(s)
eajs
sn+c
)
ds
s− z
∗ ∗ · · · ∗ ∗
...
...
. . .
...
...
∗ ∗ . . . ∗ ∗
for z ∈ C \ γ. The other rows are filled with type I multiple orthogonal polynomials of slightly
different degrees (for details, see [20]).
One can reduce the size of the Riemann–Hilbert problem as follows. Divide both sides of (2.6)
by W and carry over Pn to the right-hand side. Then, we obtain
p∑
j=1
Qj(z)
eajz
W (z)
= −Pn(z) +O
(
zn+c
)
, (5.2)
where we assume that aj ̸= 0 for all j. Then by Cauchy’s formula we obtain
1
2πi
∮
γ
p∑
j=1
Qj(s)
eajs
sn+cW (s)
sk ds = −δk,c−1, k = 0, . . . , c− 1, (5.3)
where γ is a counterclockwise-oriented closed contour going around the origin once, such that
all the zeros of W lie in its exterior. This formulation of the type I multiple orthogonality leads
to a Riemann–Hilbert problem of size (p+ 1)× (p+ 1).
Note that we use Y again to denote the solution of the Riemann–Hilbert problem, although
this solution is different from the earlier one that we also called Y . We trust that this does not
lead to any confusion.
Riemann–Hilbert problem (type I multiple orthogonality for Q1, . . . , Qp). Find solution
Y : C \ γ → C(p+1)×(p+1) such that
(1) Y (z) is analytic for z ∈ C \ γ;
(2) Y+(z) = Y−(z)JY (z) for z ∈ γ, where JY (z) =
1 · · · 0 ea1z/(zn+cW (z))
...
. . .
...
...
0 · · · 1 eapz/(zn+cW (z))
0 · · · 0 1
;
(3) Y (z) =
(
I +O
(
1
z
))
zc1 · · · · · · 0
...
. . .
...
...
0 · · · zcp 0
0 · · · 0 z−c
as z → ∞.
The unique solution to this Riemann–Hilbert problem has the polynomials Q1, . . . , Qp in its
last row,
Y (z) =
∗ · · · ∗ ∗
...
. . .
...
...
∗ · · · ∗ ∗
Q1(z) · · · Qp(z)
1
2πi
∮
γ
p∑
j=1
Qj(s)
eajs
sn+cW (s)
ds
s− z
,
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 13
for z ∈ C \ γ. Once we know Q1, . . . , Qp, then Pn can be recovered from (5.2). That is, −Pn is
the n-th partial sum of the Maclaurin series of
∑p
j=1Qj(z)
eajz
W (z) .
As a remark we add that if a1 = 0, then the O-term in (5.2) changes to O
(
zn+c−c1
)
. The
condition (5.3) and the Riemann–Hilbert problem similar to the above can still be written after
appropriate modifications.
In a similar manner, we can single out one of the Qj in (2.6). For the ease of notation, let us
choose Qp and rewrite (2.6) as
Pn(z)W (z)e−apz +
p−1∑
j=1
Qj(z)e
(aj−ap)z = −Qp(z) +O
(
zn+c
)
.
Then, by Cauchy’s integral formula,
1
2πi
∮
γ
(
Pn(s)
W (s)e−aps
sn+c
+
p−1∑
j=1
Qj(s)
e(aj−ap)s
sn+c
)
sk ds = 0 (5.4)
with k = 0, . . . , n+ c− cp − 1, where γ is an arbitrary closed contour around the origin.
The corresponding Riemann–Hilbert problem is as follows.
Riemann–Hilbert problem (type I multiple orthogonality for Pn, Q1, . . . , Qp−1). Find solu-
tion Y : C \ γ → C(p+1)×(p+1) such that
(1) Y (z) is analytic for z ∈ C \ γ;
(2) Y+(z) = Y−(z)JY (z), z ∈ γ,
1 0 · · · 0 W (z)e−apz/zn+c
0 1 · · · 0 e(a1−ap)z/zn+c
...
...
. . .
...
...
0 0 · · · 1 e(ap−1−ap)z/zn+c
0 0 · · · 0 1
;
(3) Y (z) =
(
I +O
(
1
z
))
zn 0 · · · 0 0
0 zc1 · · · 0 0
...
...
. . .
...
...
0 0 · · · zcp−1 0
0 0 · · · 0 z−n−c+cp
as z → ∞.
The unique solution is
Y (z) =
Pn(z) Q1(z) · · · Qp−1(z)
1
2πi
∮
γ
(
Pn(s)
W (s)e−apz
sn+c
+
p−1∑
j=1
Qj(s)
e(aj−ap)s
sn+c
)
ds
s−z
∗ ∗ · · · ∗ ∗
...
...
. . .
...
...
∗ ∗ · · · ∗ ∗
for z ∈ C \ γ.
For p = 1, we recover the Riemann–Hilbert problem for the orthogonal polynomials from [3].
Indeed, if p = 1, W (z) = (z − a)c, then (5.4) yields∫
γ
Pn(s)W (s)e−as
sn+c
sk ds = 0, k = 0, . . . , n− 1,
14 S. Berezin, A.B.J. Kuijlaars and I. Parra
which is a usual non-Hermitian orthogonality on a contour. The above Riemann–Hilbert problem
reduces to the usual Riemann–Hilbert problem for orthogonal polynomials that is known from
the work of Balogh, Bertola, Lee, and McLaughlin [3]. The first row of the solution is
Y (z) =
Pn(z)
1
2πi
∫
γ
Pn(s)(s− a)ce−as
sn+c
ds
s− z
∗ ∗
, z ∈ C \ γ.
6 Type II multiple orthogonality
The planar orthogonality corresponding to (1.1)–(1.2) is equivalent to the type II multiple or-
thogonality. This has been established by Lee and Yang in [17]. We give a proof in the case of
polynomial W , which is essentially the same as the one in [17], except that the situation is more
transparent due to the lack of branch cuts for W and ϕk.
Type II multiple orthogonality means that there exists functions w1, . . . , wp and non-negative
integers n1, . . . , np with n =
∑p
k=1 nk such that Pn is the unique polynomial of degree n satisfying
1
2πi
∮
γ
Pn(z)W (z)zmwk(z) dz = 0 (6.1)
for k = 1, . . . , p and m = 0, . . . , nk − 1, where γ is a closed contour around the origin.
Theorem 6.1. Suppose W is a polynomial. Given n ∈ N, choose integers n1, . . . , np such that
p∑
j=1
nj = n,
⌊
n
p
⌋
≤ nj ≤
⌈
n
p
⌉
, (6.2)
and define wk, k = 1, . . . , p, by
wk(z) =
∫ z̄×∞
0
p∏
j=1
(u− aj)
cj+nj−δk,je−uz du. (6.3)
Then, the planar orthogonal polynomial Pn is the unique monic polynomial of degree n that
satisfies (6.1).
Note that (6.2) implies |nj − nk| ≤ 1 for every j, k = 1, . . . , p. Euclid’s division lemma yields
n = ap+ b
for some a, b ∈ N ∪ {0} such that 0 ≤ b < p. And it is clear that a + 1 and a will appear
in (n1, . . . , np) exactly b and p− b times, respectively.
Proof of Theorem 6.1. Suppose that Pn is the degree n monic planar orthogonal polynomial.
Take k ∈ {1, . . . , p}, and let m be an integer such that 0 ≤ m ≤ nk − 1. Carrying out m
integration by parts in (6.3), we get
zmwk(z) = (−1)m
∫ z̄×∞
0
p∏
j=1
(u− aj)
cj+nj−δk,j
[(
d
du
)m
e−uz
]
du
=
∫ z̄×∞
0
[(
d
du
)m p∏
j=1
(u− aj)
cj+nj−δk,j
]
e−uz du+Πk,m(z), (6.4)
where Πk,m is a polynomial that comes from the boundary terms at u = 0.
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 15
Observe that(
d
du
)m p∏
j=1
(u− aj)
cj+nj−δk,j (6.5)
is a polynomial in u of degree ≤ c + n − 1 − m with a zero of order cj + nj − δk,j − m at aj
for every j = 1, . . . , p. From the definition (6.2) it follows that |nk − nj | ≤ 1, which implies
that nk ≤ nj + 1− δk,j for every j = 1, . . . , p. Using m ≤ nk − 1, we find that (6.5) has a zero
at aj of order ≥ cj . Therefore, (6.5) is divisible by W ∗(u), and we can write(
d
du
)m p∏
j=1
(u− aj)
cj+nj−δk,j = W ∗(u)Qk,m(u), (6.6)
where Qk,m is a polynomial of degree deg(Qk,m) = n− 1−m ≤ n− 1 and Qk,m has a zero at aj
of order nj − δk,j −m for every j = 1, . . . , p.
In view of (6.4) and (6.6), we then have
1
2πi
∮
γ
Pn(z)W (z)zmwk(z) dz =
1
2πi
∮
γ
Pn(z)W (z)
[∫ z̄×∞
0
W ∗(u)Qk,m(u)e−uzdu
]
dz
+
1
2πi
∮
γ
Pn(z)W (z)Πk,m(z) dz.
The second term vanishes because of Cauchy’s theorem, and the remaining term vanishes because
of part (b) in Theorem 2.1 and the fact that deg(Qk,m) ≤ n− 1. Hence, (6.1) holds.
Conversely, suppose that Pn satisfies (6.1) with wk and nk as in the statement of the theorem.
Then, by (6.4) and (6.6), we get
1
2πi
∮
γ
Pn(z)W (z)
∫ z̄×∞
0
W ∗(u)Qk,m(u)e−uz dudz = 0,
for k = 1, . . . , p andm = 0, . . . , nk−1. Again, there is no contribution from the polynomial Πk,m.
This leads directly to (2.3), provided that the Qk,m are a basis of the vector space of polynomials
of degree ≤ n − 1. Then, Theorem 2.1 tells us that Pn is the degree n planar orthogonal
polynomial.
The polynomials Qk,m for k = 1, . . . , p and m = 0, . . . , nk − 1 have degrees ≤ n − 1, and
because of (6.2) there are n of them. Thus, it suffices to prove that the Qk,m are linearly
independent. Suppose that βk,m are complex numbers such that
p∑
k=1
nk−1∑
m=0
βk,mQk,m = 0. (6.7)
We already noted in the first part of the proof that Qk,m has a zero at aj of exact order
nj − δk,j −m. Hence, Qk,m(aj) ̸= 0 if and only if nj − δk,j −m = 0.
From (6.2), we have that |nj − nk| ≤ 1, and since m ≤ nk − 1, it is then easy to see
that Qk,m(aj) ̸= 0 if and only if m = nk − 1, and either k = j, or k ̸= j and nk = nj + 1. Thus,
by evaluating (6.7) at aj , we obtain
βj,nj−1Qj,nj−1(aj) +
p∑
k=1
nk=nj+1
βk,nk−1Qk,nk−1(aj) = 0. (6.8)
16 S. Berezin, A.B.J. Kuijlaars and I. Parra
Suppose nj0 =
⌈
n
p
⌉
. Then, there are no indices k with nk = nj0 + 1, and (6.8) implies
that βj0,nj0
−1 = 0 since Qj0,nj0
−1(aj0) ̸= 0. Suppose nj0 =
⌊
n
p
⌋
. Then, every k with nk = nj0 +1
satisfies nk =
⌈
n
p
⌉
, and we just proved that βk,nk−1 = 0 for such k. Thus, by using (6.8)
again we obtain that βj0,nj0
−1 = 0. Since j0 can be arbitrary, we conclude that βj,nj−1 = 0 for
all j = 1, . . . , p.
The formula (6.8) reduces to
p∑
k=1
nk−2∑
m=0
βk,mQk,m = 0. (6.9)
We continue by looking at the remaining polynomials Qk,m in (6.9) and observe that the only
ones with d
dzQk,m
∣∣
z=aj
̸= 0 are those with m = nk−2 and either k = j, or k ̸= j and nk = nj+1.
Arguing as before we find that βj,nj−2 = 0 for every j = 1, . . . , p. Continuing in this way by
considering the higher order derivatives, we ultimately find that βk,m = 0 for all k = 1, . . . , p and
m = 0, . . . , nk − 1, which shows that the polynomials Qk,m are indeed linearly independent. ■
The connection between Riemann–Hilbert problems and type II multiple orthogonality is
well known, see Van Assche, Geronimo, and Kuijlaars [20]. Therefore, we arrive at the following
Riemann–Hilbert problem of size (p+1)× (p+1), corresponding to Theorem 6.1 (first appeared
in Lee and Yang [17]).
Riemann–Hilbert problem (type II multiple orthogonality). Let w1, . . . , wp be given by for-
mula (6.3). Find Y : C \ γ → C(p+1)×(p+1) satisfying
(1) Y (z) is analytic for z ∈ C \ γ;
(2) Y+(z) = Y−(z)JY (z) for z ∈ γ, where JY (z) =
1 w1(z) · · · wp(z)
0 1 · · · 0
...
...
. . .
...
0 0 · · · 1
;
(3) Y (z) =
(
I +O
(
1
z
))
zn 0 · · · 0
0 z−n1 · · · 0
...
...
. . .
...
0 0 · · · z−np
as z → ∞.
The unique solution has Pn in its first row
Y (z) =
Pn(z)
1
2πi
∮
γ
Pn(s)w1(s)
ds
s− z
· · · 1
2πi
∮
γ
Pn(s)wp(s)
ds
s− z
∗ ∗ · · · ∗
...
...
. . .
...
∗ ∗ · · · ∗
(6.10)
for z ∈ C \ γ.
7 Conclusion
It is known that the planar orthogonal polynomials orthogonal with respect to the measure (1.1)
are multiple orthogonal polynomials of type II (see [17]). Assuming W is a polynomial weight,
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials 17
we have shown that the same planar orthogonal polynomials are also multiple orthogonal poly-
nomials of type I. It is remarkable that the planar orthogonality manifests in two different ways
at once. We are not aware of any other examples of such a phenomenon except for the case
p = 1, where both situations reduce to the usual orthogonality.
Generally speaking, multiple orthogonality of each type, I and II, is characterized by a Rie-
mann–Hilbert problem. Moreover, there is a canonical correspondence between such Riemann–
Hilbert problems. Indeed, if Y is given by (6.10) then not only does it solve a Riemann–Hilbert
problem for multiple orthogonal polynomials of type II, but also its inverse transpose Y −t
contains multiple orthogonal polynomials of type I in each of its rows. In the course of the paper,
we have established several different Riemann–Hilbert problems connected with orthogonality
of type I. However, if p ≥ 2, then the orthogonal polynomial Pn enters the inverse transpose
of (6.10) only as part of a bigger algebraic expression and not by itself as it should in the case
of the canonical correspondence; therefore, neither of our Riemann–Hilbert problems associated
to multiple orthogonality of type I are related to the type II problem in a canonical way.
A major interest in stating the Riemann–Hilbert problems is to use them for the asymptotic
analysis. We do not address this topic in the present paper, however we mention that Lee and
Yang [18] used the Riemann–Hilbert problem corresponding to multiple orthogonality of type II
for asymptotic analysis in the situation where the cj are fixed and n → ∞. It would be very
interesting to deal with the case of varying weights, namely, when the cj depend on n and tend
to infinity at a rate proportional to n. For p = 1 this was accomplished in [3], and we hope that
one of the Riemann–Hilbert problems in our paper can be useful for the case p ≥ 2.
Acknowledgments
S.B. is supported by FWO Senior Postdoc Fellowship, project 12K1823N. A.B.J.K. was sup-
ported by the long term structural funding “Methusalem grant of the Flemish Government”,
and by FWO Flanders projects EOS 30889451 and G.0910.20. I.P. was supported by FWO
Flanders project G.0910.20.
References
[1] Akemann G., Nagao T., Parra I., Vernizzi G., Gegenbauer and other planar orthogonal polynomials on an
ellipse in the complex plane, Constr. Approx. 53 (2021), 441–478, arXiv:1905.02397.
[2] Akemann G., Vernizzi G., Characteristic polynomials of complex random matrix models, Nuclear Phys. B
660 (2003), 532–556, arXiv:hep-th/0212051.
[3] Balogh F., Bertola M., Lee S.-Y., McLaughlin K.D.T.-R., Strong asymptotics of the orthogonal polyno-
mials with respect to a measure supported on the plane, Comm. Pure Appl. Math. 68 (2015), 112–172,
arXiv:1209.6366.
[4] Balogh F., Grava T., Merzi D., Orthogonal polynomials for a class of measures with discrete rotational
symmetries in the complex plane, Constr. Approx. 46 (2017), 109–169, arXiv:1509.05331.
[5] Bertola M., Elias Rebelo J.G., Grava T., Painlevé IV critical asymptotics for orthogonal polynomials in the
complex plane, SIGMA 14 (2018), 091, 34 pages, arXiv:1802.01153.
[6] Bleher P.M., Kuijlaars A.B.J., Orthogonal polynomials in the normal matrix model with a cubic potential,
Adv. Math. 230 (2012), 1272–1321, arXiv:1106.6168.
[7] Byun S.-S., Forrester P.J., Progress on the study of the Ginibre ensembles I: GinUE, arXiv:2211.16223.
[8] Deaño A., Simm N., Characteristic polynomials of complex random matrices and Painlevé transcendents,
Int. Math. Res. Not. 2022 (2022), 210–264, arXiv:1909.06334.
[9] Deift P., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lect. Notes
Math., Vol. 3, New York University, New York, 1999.
[10] Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Uniform asymptotics for polyno-
mials orthogonal with respect to varying exponential weights and applications to universality questions in
random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425.
https://doi.org/10.1007/s00365-020-09515-0
https://arxiv.org/abs/1905.02397
https://doi.org/10.1016/S0550-3213(03)00221-9
https://arxiv.org/abs/hep-th/0212051
https://doi.org/10.1002/cpa.21541
https://arxiv.org/abs/1209.6366
https://doi.org/10.1007/s00365-016-9356-0
https://arxiv.org/abs/1509.05331
https://doi.org/10.3842/SIGMA.2018.091
https://arxiv.org/abs/1802.01153
https://doi.org/10.1016/j.aim.2012.03.021
https://arxiv.org/abs/1106.6168
https://arxiv.org/abs/2211.16223
https://doi.org/10.1093/imrn/rnaa111
https://arxiv.org/abs/1909.06334
https://doi.org/10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
18 S. Berezin, A.B.J. Kuijlaars and I. Parra
[11] Deift P., Zhou X., A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for
the MKdV equation, Ann. of Math. 137 (1993), 295–368, arXiv:math.AP/9201261.
[12] Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in 2D quantum gravity,
Comm. Math. Phys. 147 (1992), 395–430.
[13] Gustafsson B., Teodorescu R., Vasil’ev A., Classical and stochastic Laplacian growth, Adv. Math. Fluid
Mech., Birkhäuser, Cham, 2014.
[14] Hedenmalm H., Wennman A., Planar orthogonal polynomials and boundary universality in the random
normal matrix model, Acta Math. 227 (2021), 309–406, arXiv:1710.06493, .
[15] Karp D., Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, in
Analytic Extension Formulas and Their Applications (Fukuoka, 1999/Kyoto, 2000), Int. Soc. Anal. Appl.
Comput., Vol. 9, Kluwer Acad. Publ., Dordrecht, 2001, 169–187.
[16] Lee S.-Y., Yang M., Discontinuity in the asymptotic behavior of planar orthogonal polynomials under
a perturbation of the Gaussian weight, Comm. Math. Phys. 355 (2017), 303–338, arXiv:1607.02821.
[17] Lee S.-Y., Yang M., Planar orthogonal polynomials as Type II multiple orthogonal polynomials, J. Phys. A
52 (2019), 275202, 14 pages, arXiv:1801.01084.
[18] Lee S.-Y., Yang M., Strong asymptotics of planar orthogonal polynomials: Gaussian weight perturbed by
finite number of point charges, arXiv:2003.04401.
[19] Van Assche W., Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equa-
tions, in Orthogonal Polynomials, Tutor. Sch. Workshops Math. Sci., Birkhäuser, Cham, 2020, 629–683,
arXiv:1904.07518.
[20] Van Assche W., Geronimo J.S., Kuijlaars A.B.J., Riemann–Hilbert problems for multiple orthogonal poly-
nomials, in Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci.
Ser. II Math. Phys. Chem., Vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, 23–59.
[21] van Eijndhoven S.J.L., Meyers J.L.H., New orthogonality relations for the Hermite polynomials and related
Hilbert spaces, J. Math. Anal. Appl. 146 (1990), 89–98.
[22] Webb C., Wong M.D., On the moments of the characteristic polynomial of a Ginibre random matrix, Proc.
Lond. Math. Soc. 118 (2019), 1017–1056, arXiv:1704.04102.
https://doi.org/10.2307/2946540
https://arxiv.org/abs/math.AP/9201261
https://doi.org/10.1007/BF02096594
https://doi.org/10.1007/978-3-319-08287-5
https://arxiv.org/abs/#2
https://doi.org/10.4310/acta.2021.v227.n2.a3
https://doi.org/10.1007/978-1-4757-3298-6_10
https://doi.org/10.1007/s00220-017-2888-8
https://arxiv.org/abs/1607.02821
https://doi.org/10.1088/1751-8121/ab1af9
https://arxiv.org/abs/1801.01084
https://arxiv.org/abs/2003.04401
https://doi.org/10.1007/978-3-030-36744-2_22
https://arxiv.org/abs/1904.07518
https://doi.org/10.1007/978-94-010-0818-1_2
https://doi.org/10.1016/0022-247X(90)90334-C
https://doi.org/10.1112/plms.12225
https://doi.org/10.1112/plms.12225
https://arxiv.org/abs/1704.04102
1 Introduction
2 Statement of result
3 Fundamental identity
4 Proof of Theorem 2.1
5 Type I multiple orthogonality
6 Type II multiple orthogonality
7 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-211864 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T07:04:37Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Berezin, Sergey Kuijlaars, Arno B.J. Parra, Iván 2026-01-14T09:52:20Z 2023 Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials. Sergey Berezin, Arno B.J. Kuijlaars and Iván Parra. SIGMA 19 (2023), 020, 18 pages 1815-0659 2020 Mathematics Subject Classification: 42C05; 30E25; 41A21 arXiv:2212.06526 https://nasplib.isofts.kiev.ua/handle/123456789/211864 https://doi.org/10.3842/SIGMA.2023.020 A recent result of S.-Y. Lee and M. Yang state that the planar orthogonal polynomials orthogonal with respect to a modified Gaussian measure are multiple orthogonal polynomials of type II on a contour in the complex plane. We show that the same polynomials are also type I orthogonal polynomials on a contour, provided the exponents in the weight are integers. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based on the fundamental identity of Lee and Yang, which we establish using a new technique. S.B. is supported by FWO Senior Postdoc Fellowship, project 12K1823N. A.B.J.K. was supported by the long-term structural funding “Methusalem grant of the Flemish Government”, and by FWO Flanders projects EOS 30889451 and G.0910.20. I.P. was supported by FWO Flanders project G.0910.20. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials Article published earlier |
| spellingShingle | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials Berezin, Sergey Kuijlaars, Arno B.J. Parra, Iván |
| title | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials |
| title_full | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials |
| title_fullStr | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials |
| title_full_unstemmed | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials |
| title_short | Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials |
| title_sort | planar orthogonal polynomials as type i multiple orthogonal polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211864 |
| work_keys_str_mv | AT berezinsergey planarorthogonalpolynomialsastypeimultipleorthogonalpolynomials AT kuijlaarsarnobj planarorthogonalpolynomialsastypeimultipleorthogonalpolynomials AT parraivan planarorthogonalpolynomialsastypeimultipleorthogonalpolynomials |