A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type
We present a representation of skew-orthogonal polynomials of symplectic type ( = 4) in terms of a matrix Riemann-Hilbert problem, for weights of the form e⁻ⱽ⁽ᶻ⁾ where is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multi...
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| description | We present a representation of skew-orthogonal polynomials of symplectic type ( = 4) in terms of a matrix Riemann-Hilbert problem, for weights of the form e⁻ⱽ⁽ᶻ⁾ where is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a = 4 analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a = 4 analogue of the Toda lattice.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 076, 32 pages
A Riemann–Hilbert Approach to Skew-Orthogonal
Polynomials of Symplectic Type
Alex LITTLE
Unité de Mathématiques Pures et Appliquées, ENS de Lyon, France
E-mail: alexander.little@ens-lyon.fr
Received December 27, 2023, in final form August 06, 2024; Published online August 16, 2024
https://doi.org/10.3842/SIGMA.2024.076
Abstract. We present a representation of skew-orthogonal polynomials of symplectic
type (β = 4) in terms of a matrix Riemann–Hilbert problem, for weights of the form e−V (z)
where V is a polynomial of even degree and positive leading coefficient. This is done by
representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive
a β = 4 analogue of the Christoffel–Darboux formula. Finally, our Riemann–Hilbert repre-
sentation allows us to derive a Lax pair whose compatibility condition may be viewed as
a β = 4 analogue of the Toda lattice.
Key words: Riemann–Hilbert problem; skew-orthogonal polynomials; random matrices
2020 Mathematics Subject Classification: 60B20; 33C45; 30E15; 30E25
1 Introduction
In this paper we develop a Riemann–Hilbert approach to the theory of skew-orthogonal poly-
nomials of symplectic type. In order to motivate our work, let us recall the basic theory of
orthogonal polynomials. One begins with a measurable function w : R −→ [0,+∞] with fi-
nite moments to all orders, that is
∫
R |x|kw(x)dx < +∞ for all k ∈ N. Then one constructs
a Hermitian form
⟨P,Q⟩2 =
∫
R
P (x)Q(x)w(x)dx
for (real) polynomials P and Q. We would like that ⟨·, ·⟩2 be nondegenerate on the linear space
of real polynomials so that we can make it into an inner product, which would be true if w is
continuous and has positive mass
∫
Rwdx > 0. Given such an inner product there is a unique
sequence of polynomials {Pn}n∈N such that Pn(x) = xn+O
(
xn−1
)
and ⟨Pn, Pm⟩2 = 0 for n ̸= m,
constructable by a Gram–Schmidt procedure. w is referred to as the weight function and Pn is
called the n-th monic orthogonal polynomial with respect to w.
The applications of orthogonal polynomials are extremely diverse, however in this work we
shall be interested solely in their application to random matrix theory (see, e.g., [14]). Let M
be an n× n Hermitian random matrix distributed according to a probability measure
1
Zn
e− trV (M)dM, (1.1)
where dM =
∏n
k=1 dMkk
∏
1≤i<j≤n dRe(Mij)d Im(Mij), V is a real-valued analytic function
on R and Zn> 0 is a constant of normalisation. For this measure to be normalisable, V (x) → +∞
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:alexander.little@ens-lyon.fr
https://doi.org/10.3842/SIGMA.2024.076
https://www.emis.de/journals/SIGMA/Its.html
2 A. Little
as x→ ±∞ sufficiently rapidly, more precisely, lim infx→±∞
V (x)
log |x| = +∞. This condition would
be satisfied if V was a polynomial of even degree and positive leading coefficient, as will be the
case in this paper.
A remarkable calculation (see [35, Chapter 5] or [28, Chapter 5]) then shows that the corre-
lation functions of the eigenvalues of such an ensemble are expressible entirely in terms of the
orthogonal polynomials Pn and Pn−1 and their respective norms, ∥Pn∥ and ∥Pn−1∥, with respect
to the weight function w(x) = e−V (x). Thus to make an asymptotic analysis of the correlation
functions as n → ∞ it is necessary to make an asymptotic analysis of orthogonal polynomials
with respect to more or less arbitrary weight functions. For certain “classical” weight functions,
this can be done by classical steepest descent techniques, however the general case required
the discovery of a Riemann–Hilbert representation of orthogonal polynomials [27] and the de-
velopment of the nonlinear steepest descent technique [23]. A Riemann–Hilbert representation
involves representing an object of interest in terms of the solution to a boundary value problem
in the complex plane with prescribed behaviour at infinity, and can be regarded as a gener-
alisation of the notion of a contour integral [30]. This led to a proof of universality of local
correlations in the bulk and at the edge of the spectrum for ensembles of type (1.1) by means of
the so-called Deift–McLaughlin–Kriecherbauer–Venakides–Zhou (DMKVZ) scheme [20, 21, 22],
building off earlier work of Pastur–Shcherbina [37] and Bleher–Its [6].
Motivated by, amongst others, applications to quantum mechanical systems with time-rever-
sal invariance (see [35, Chapter 2]), random matrix theorists have also been interested for a long
time in studying not only Hermitian but also real symmetric and quaternion self-dual random
matrices with probability measures of the form of (1.1). These three cases, real symmetric, com-
plex Hermitian and quaternion self-dual, correspond to the three finite-dimensional, associative
division algebras over the real numbers, R, C and H, respectively, [24]. These are often labelled
by the “Dyson index” β = 1, 2, 4, respectively.
In this work, we shall be interested in the case of β = 4, the so-called symplectic case, since
the probability measure is invariant under conjugation by elements of the compact symplectic
group, and since we prefer to think of an n × n quaternion matrix as a 2n × 2n matrix where
the quaternions are represented by 2 × 2 blocks. For such a 2n × 2n matrix, there are, with
probability 1, only n distinct eigenvalues since every eigenvalue occurs with multiplicity 2 (known
in the physics literature as Kramers’ degeneracy). These n eigenvalues have the joint PDF
ρn(x1, . . . , xn) =
1
Z̃n
∏
1≤i<j≤n
|xi − xj |4
n∏
k=1
e−2V (xk), (1.2)
where Z̃n is a normalisation constant. We shall furthermore restrict to the case of V being a real
polynomial of even degree D and leading coefficient γ > 0,
V (x) = γxD +O
(
xD−1
)
, x→ ∞.
Here the classical approach (see [35, Chapter 5], [28, Chapter 6]) involves finding an analogue
of orthogonal polynomials and this is supplied by skew-orthogonal polynomials. Here the inner
product ⟨·, ·⟩2 is replaced by a skew-symmetric nondegenerate bilinear form ⟨·, ·⟩4, where
⟨P,Q⟩4
def
=
1
2
∫
R
(P (x)Q′(x)− P ′(x)Q(x))e−2V (x)dx (1.3)
for P and Q real polynomials. By “non-degenerate”, we mean the following. Let PR
k be the
space of real polynomials of degree ≤ k. Then we say ⟨·, ·⟩4 is non-degenerate if for all n ∈ N,
we have the implication
∀P ∈ PR
2n+1
(
⟨P,Q⟩4 = 0, ∀Q ∈ PR
2n+1 =⇒ P = 0
)
.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 3
Then a sequence of monic polynomials {Pn}n∈N, Pn(x) = xn +O
(
xn−1
)
, is said to be skew-
orthogonal with respect to ⟨·, ·⟩4 if the following holds:
(1) ⟨P2n, P2m⟩4 = ⟨P2n+1, P2m+1⟩4 = 0 for all n,m ∈ N.
(2) ⟨P2n, P2m+1⟩4 = 0 for all n,m ∈ N, n ̸= m.
The existence of such a sequence is guaranteed by non-degeneracy (it can be constructed by
a procedure analogous to the Gram–Schmidt method), however these properties do not uniquely
define the sequence because one can replace P2n+1 7→ P2n+1 + αP2n for any constant α ∈ R.
This degree of freedom may be fixed by demanding that the next-to-leading coefficient of P2n+1
is equal to a specified value (it is common to choose this value to be zero however this will not
be the most convenient choice for us). It can then be verified that with this extra constraint the
sequence is unique.
The importance of skew-orthogonal polynomials for random matrix theory can be seen by the
following. Let hk = ⟨P2k, P2k+1⟩4 be the “skew norm”, where by the non-degeneracy of ⟨·, ·⟩4 we
necessarily have hk ̸= 0. Then (see [28, Propositions 6.1.6 and 6.1.7]) all eigenvalue correlation
functions associated to the point process (1.2) may be expressed in terms of the so-called pre-
kernel,
Sn(x, y)
def
=
n−1∑
k=0
P2k(x)e
−V (x) d
dy
(
P2k+1(y)e
−V (y)
)
− P2k+1(x)e
−V (x) d
dy
(
P2k(y)e
−V (y)
)
2hk
. (1.4)
Precisely, from Sn one constructs a 2 × 2 matrix “kernel”, and then the k-point correla-
tion function of the ensemble is written as a 2k × 2k Pfaffian, or equivalently, quaternion
determinant (see [28, Chapter 6]). In particular, the 1-point correlation function is given
by ρ
(1)
n (x) = Sn(x, x). A similar treatment in terms of skew-orthogonal polynomials, though
with a different skew-inner product than (1.3), can be done in the case of real symmetric ma-
trices (β = 1).
The importance of understanding skew-orthogonal polynomials of real and symplectic type
(β = 1, 4) was stressed by Dyson [25] when he wrote:
We need to develop the theory of skew-orthogonal polynomials until it becomes
a working tool as handy as the existing theory of orthogonal polynomials.
Unfortunately, this theory remains comparatively underdeveloped relative to that of orthogonal
polynomials (β = 2). Nevertheless let us state some known results. Adler, Horozov and van Mo-
erbeke [2] give Pfaffian formulas for the even and odd skew-orthogonal polynomials for β = 1, 4.
Eynard [26] derives Heine-type multiple integral formulas for even and odd skew-orthogonal
polynomials for β = 1, 4 and he uses a non-rigorous but intriguing resolvent method to ob-
tain n→ ∞ asymptotics for the case of a one-cut polynomial potential. These multiple-integral
formulas are morally equivalent to the Pfaffian formulas since the two may be related by de
Bruijn identities [13]. In the case of V ′ being rational, Adler and van Moerbeke [3] find a nat-
ural basis of orthogonal polynomials in which to express the corresponding skew-orthogonal
polynomials and Adler, Forrester, Nagao and van Moerbeke [1] use this to write simple ex-
pressions for the skew-orthogonal polynomials in the classical (Hermite, Laguerre and Jacobi)
cases.
A key inspiration for the present paper is the paper of Pierce [38] in which a Riemann–Hilbert
representation for skew-orthogonal polynomials of real type (β = 1) with a polynomial V is
derived; unfortunately this work has not received nearly the same attention as the corresponding
problem for orthogonal polynomials because there is no known Christoffel–Darboux-type formula
to express the pre-kernel in terms of the Riemann–Hilbert problem, and even if there was, the
complexity of the jump matrix makes it unclear how to perform a nonlinear steepest descent
4 A. Little
analysis. Pierce’s paper is similar to the present one in that it considers polynomial potentials
and this allows one to represent (β = 1) skew-orthogonality as a kind of multiple-orthogonality,
and this naturally leads to a Riemann–Hilbert problem.
Although skew-orthogonal polynomials on the real line have received relatively little recent
attention, there has been much recent work on planar symplectic ensembles and their associ-
ated skew-orthogonal polynomials. Akemann, Ebke and Parra [4] give a construction of skew-
orthogonal polynomials in the plane and relate these to a corresponding set of orthogonal poly-
nomials; while several recent works [8, 9, 10] derive a “generalised Christoffel–Darboux formula”
for several different ensembles, which is a differential equation relating the kernel for the sym-
plectic ensemble to that of the corresponding complex ensemble. These works are in a similar
spirit to the method of Widom [44] in that they establish relations between the symplectic en-
semble and the corresponding complex ensemble, the latter of which is often more tractable.
This should be contrasted with the present work where our Riemann–Hilbert formulation does
not make any direct connection with the corresponding random matrix ensemble for β = 2.
The difficulty of working with skew-orthogonal polynomials has led some to believe that they
are the wrong way to approach β = 1, 4 ensembles and to develop a different formalism that
avoids skew-orthogonal polynomials entirely. This is the Widom formalism [44], presented and
developed in the excellent book of Deift and Gioev [18], and it is this formalism that has yielded
proofs of universality of local correlations for β = 1, 4 [16, 17, 19, 31, 40, 39, 41, 42]. One of
the aims of this paper is to show that an approach involving skew-orthogonal polynomials can
nevertheless be highly fruitful and that it is possible to treat β = 4 ensembles in a self-contained
way, without making a reduction to β = 2.
Let us now state our results. Given the sequence of skew-orthogonal polynomials {Pn}n∈N,
we define the “dual” polynomials as Ψn(x)
def
= − 1
γDeV (x) d
dx
(
Pn(x)e
−V (x)
)
.
(1) In Section 2, we show how Ψ2n and Ψ2n+1 may be thought of as “multiple-orthogonal
polynomials of Type II” (see Corollary 2.23). This yields two families of Riemann–Hilbert
Problems 2.7 and 2.13 (An and Bn, respectively), where Ψ2n and Ψ2n+1 appear as the
(1, 1) matrix element of An and Bn, respectively. An has dimension D + 1 and Bn has
dimension D + 2. Furthermore, we find that P2n is the (2, 2) matrix element of A−1
n ,
however, somewhat contrary to what one might expect, neither P2n+1 nor any multiple of
it appears as a matrix element of B−1
n . This seems to be related to the fact that P2n can
be thought of as a multiple-orthogonal polynomial of Type I (see Corollary 2.25), but we
are not aware of a way to represent P2n+1 in such a way.
(2) In Section 3, we prove an analogue of the Christoffel–Darboux formula, namely
Sn(x, y) = − 1
4πi
e−V (x)−V (y)
(
A−1
n (x)An(y)
)
21
x− y
, (1.5)
where Sn is the pre-kernel (1.4) and An is the solution of Riemann–Hilbert Problem 2.7. If
a nonlinear steepest descent analysis could be carried out for An, we would thus obtain a
new proof of universality for β = 4, perhaps easier than existing proofs. It is also interesting
to remark that the pre-kernel is expressible entirely in terms of the “even” Riemann–Hilbert
problem An, i.e., from the point of view of random matrix theory the odd problem Bn
can be ignored. This is not the first instance that an analogue of the Christoffel–Darboux
formula for skew-orthogonal polynomials has appeared in the literature; in [29], Ghosh
derives a formula for (x−y)Sn(x, y) in terms of something which, for polynomial potentials,
has finite rank. However, no connection to a Riemann–Hilbert problem was made so an
asymptotic analysis of this formula was limited to special cases.
(3) In Section 4, we show how the An problem may be transformed to have constant jump
matrices. Then allowing the potential V to depend on a parameter t ∈ R by V (z, t) =
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 5
V0(z)+zt, we derive an (n, t) Lax pair. The linear system in n could be regarded as a β = 4
analogue of the “three term recurrence” which appears in the theory of orthogonal poly-
nomials (see Remark 4.7). The compatibility condition of this Lax pair yields a dynamical
system which one could regard as a β = 4 analogue of the Toda lattice (see Theorem 4.1).
In the D = 2 case an exact solution can be given (see Example 4.4). We should remark
that an analogue of the Toda lattice for β = 1, 4, with connections to skew-orthogonal
polynomials, already exists in the literature under the name of the “Pfaff lattice” [2]. The
relationship between the Pfaff lattice and our dynamical system is left as a topic for future
investigation.
As already remarked, if a nonlinear steepest descent analysis of An could be carried out then (1.5)
would give access to the asymptotic behaviour of many observables that were either out of reach
of previous techniques or could only be approached by cumbersome methods. For example, it
could be used to probe non-standard universality classes, analogous to those that have been
studied for β = 2 when two cuts in the support of the equilibrium measure merge [7, 11, 12] and
also in planar ensembles when one has a merging singularity [32]. Such non-standard universality
classes are much less well-understood in the β = 1, 4 cases as compared to β = 2.
1.1 Notation
(1) We shall use the notation X k to stand for the rational function X k(x) = xk for k ∈ Z.
(2) For a, b ∈ Z, we let [[a, b]] be the integer interval from a to b, [[a, b]] = [a, b] ∩ Z.
(3) For k ∈ N, define the space of real polynomials of degree at most k as PR
k , and let
PR =
⋃
k≥0 PR
k be the space of all real polynomials. Similarly, let PC
k be the space of
polynomials with complex coefficients of degree ≤ k and PC =
⋃
k≥0 PC
k .
(4) For m a positive integer and F a field (in our case always R or C), we let Mm(F) be the
space of m×m matrices with entries in F.
(5) We use the convention that 0 ∈ N.
(6) We denote the 1-torus as T = {z ∈ C : |z| = 1}.
(7) We let χA be the indicator function of the set A, i.e., χA(x) = 1 if x ∈ A and χA(x) = 0
otherwise.
(8) Let Γ ⊂ C be a piecewise smooth contour in C. Then the Cauchy transform of the
measurable function f : Γ −→ C is defined as
CΓ(f)(z) =
1
2πi
∫
Γ
f(x)
x− z
dx, z ∈ C \ Γ,
whenever this integral exists. In our case, we will always consider sufficiently “nice”
functions f and contours Γ such that the Cauchy transform is well-defined and ana-
lytic for all z ∈ C \ Γ. Furthermore, we shall always consider functions with finite mo-
ments
∫
Γ |x|
k|f(x)||dx| < +∞ for all k ≥ 0 and for which CΓ(f)(z) admits an asymptotic
expansion in 1
z as z → ∞ in the sense of Poincaré. More precisely, for every K ∈ N,
we have
CΓ(f)(z) = − 1
2πi
K∑
k=0
z−k−1
∫
Γ
xkf(x)dx+
z−K−1
2πi
∫
Γ
xK+1f(x)
x− z
dx.
If dist(z,Γ) ≥ ϵ > 0, we find that
∫
Γ
xK+1f(x)
x−z dx is bounded. However, if f is analytic
we may deform the contour to achieve boundedness in a full neighbourhood of (complex)
6 A. Little
infinity. This will be the case in all the Cauchy integrals that appear in this paper. The
general theory of such Cauchy integrals for functions of varying degrees of decay and
regularity is discussed in notes of Deift [15] and the book of Muskhelishvilli [36].
2 Riemann–Hilbert representations
Before proceeding, let us demonstrate that the problem we are considering is well-posed. The fol-
lowing proposition is of course known to experts but we include a proof to keep the presentation
complete.
Proposition 2.1. The skew-symmetric bilinear form defined by (1.3) is non-degenerate. Fur-
thermore, the skew-norms are all positive, i.e., hn = ⟨P2n, P2n+1⟩4 > 0, for all n ≥ 0.
Proof. By one of de Bruijn’s identities [13], we have
pf
(〈
X i−1,X j−1
〉
4
)2n
i,j=1
=
1
n!
∫
Rn
∏
1≤i<j≤n
|xi − xj |4
n∏
k=1
e−2V (xk)dx1 · · · dxn > 0,
where pf is the Pfaffian. Hence the matrix
(〈
X i−1,X j−1
〉
4
)2n
i,j=1
is invertible for all n ≥ 1 and so
defines a non-degenerate bilinear form. Furthermore, by performing row and column operations,
0 < pf
(〈
X i−1,X j−1
〉
4
)2n
i,j=1
= h0 · · ·hn−1, ∀n ≥ 1,
which shows that hk > 0 for all k ≥ 0. ■
We begin by observing that the symplectic inner product (1.3) can be re-written as
⟨P,Q⟩4 =
∫
R
P (x)e−V (x) d
dx
(
Q(x)e−V (x)
)
dx, P,Q ∈ PR.
Definition 2.2 (dual skew-orthogonal polynomials). Given a sequence of skew-orthogonal poly-
nomials {Pn}n∈N, let us introduce the sequence of “dual” polynomials {Ψn}n∈N, where
Ψn(x)
def
= − 1
γD
eV (x) d
dx
(
e−V (x)Pn(x)
)
= xn+D−1 +O
(
xn+D−2
)
, x→ ∞.
The monic skew-orthogonal polynomials can be reconstructed from the dual polynomials by
integration,
Pn(x) = γDeV (x)
∫ +∞
x
e−V (y)Ψn(y)dy.
In order to characterise Ψn, let us ask the following question. Given a polynomial Q ∈ PC,
under what conditions will
eV (x)
∫ +∞
x
e−V (y)Q(y)dy
also be a polynomial? Note that this is equivalent to the question of whether Q can be written
Q(x) = eV (x) d
dx
(
P (x)e−V (x)
)
for some polynomial P . Let us now proceed to answer this question.
Remark 2.3. Note that the map P 7→ eV d
dx
(
P e−V
)
which maps PC → PC is linear and
injective. Thus if Q = eV d
dx
(
P e−V
)
then this P is unique.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 7
C0
C2
C1
C4
C6
C5
C3
C7
Figure 1. This figure depicts the contours C0, . . . , CD−1 in the case D = 8. The arrows denote the
orientation of the contours. Note the outgoing orientation of C0.
Consider the contours
Ck
def
= e2πi
k
D [0,+∞), k ∈ [[0, D − 1]]. (2.1)
Let us orient these contours so C0 = [0,+∞) has left to right orientation, whereas the remaining
contours C1, . . . , CD−1 have orientation towards the origin. Then let
Γk
def
= C0 ∪ Ck, k ∈ [[1, D − 1]]. (2.2)
For example, when k = D
2 (recall D is even and positive), Γk = R with standard left to right
orientation. These contours are depicted in Figure 1 for the case D = 8.
Proposition 2.4. Let Q ∈ PC be a polynomial, then Q can written as
Q(x) = eV (x) d
dx
(
e−V (x)P (x)
)
= P ′(x)− V ′(x)P (x)
for some polynomial P ∈ PC if and only if the following integrals vanish:∫
Γk
Q(x)e−V (x)dx = 0, ∀k ∈ [[1, D − 1]] (2.3)
Proof. “Only if” is trivial since for z ∈ Ck\{0}, zD > 0, hence P (z)e−V (z) is rapidly decreasing,
hence∫
Γk
Q(x)e−V (x)dz =
∫
Γk
d
dx
(
P (x)e−V (x)
)
dx = 0, ∀k ∈ [[1, D − 1]].
For the converse claim, suppose the integrals (2.3) all vanish. Let us then divideD into degQ+ 1
with remainder, so that degQ+1 = Dℓ+r, where ℓ, r ∈ N and r ≤ D−1. We will construct a cer-
tain polynomial R as follows. If degQ ≤ D−2, then R := Q, so we may assume degQ ≥ D − 1.
We can then factorise
Q(x) = F1(x)V
′(x) +G1(x),
8 A. Little
where F1 ∈ PC has degree degQ−D+1 and G1 ∈ PC has degree at most D− 2. We can then
repeat the procedure,
F ′
1(x) = F2(x)V
′(x) +G2(x), F ′
2(x) = F3(x)V
′(x) +G3(x), . . . ,
F ′
ℓ−1(x) = Fℓ(x)V
′(x) +Gℓ(x).
Fj ∈ PC has degree degQ−Dj+1 and Gj ∈ PC has degree at most D− 2 for j = 1, . . . , ℓ. In
particular Fℓ has degree r and so the process terminates at this stage. Let
R(x) := G1(x) + · · ·+Gℓ(x) + F ′
ℓ(x).
R has degree at most D − 2.
By repeated integration by parts, we see that∫
Γk
R(x)e−V (x)dx = 0, ∀k ∈ [[1, D − 1]].
We now claim that R ≡ 0. Consider the function
C(z)
def
= eV (z)
∫ +∞
z
R(x)e−V (x)dx,
where the integration path is from z ∈ C to 0 then from 0 to +∞. C is clearly an entire function.
We claim that as |z| → +∞, C(z) → 0, uniformly in arg z. Hence by Liouville’s theorem C ≡ 0,
and hence R ≡ 0.
From our assumptions,
∫ +∞
z
R(x)e−V (x)dx =
∫ (+∞)·e2πi k
D
z
R(x)e−V (x)dx, ∀k ∈ [[1, D − 1]].
Now introduce the function φ(z) = Ṽ (z)
1
D
(
where Ṽ (z) = 1
γV (z)
)
which is holomorphic and
injective so long as |z| is sufficiently large, i.e., on a neighbourhood of (complex) infinity. Let us
choose k ∈ {0, . . . , D − 1}, so that
2π
k − 1
2
D
≤ argφ(z) ≤ 2π
k + 1
2
D
(this k may not be unique but this does not matter). Let us also define arg so that there is no
branch cut through this sector. Finally, let us choose the contour such that Re(V (z)) is always
non-decreasing along the contour. The existence of such a contour is proven in Lemma 2.5.
Integrating by parts, we find
∫ (+∞)·e2πi k
D
z
R(x)e−V (x)dx =
R(z)
V ′(z)
e−V (z) +
∫ (+∞)·e2πi k
D
z
d
dx
(
R(x)
V ′(x)
)
e−V (x)dx.
d
dx
( R(x)
V ′(x)
)
is a rational function which scales like x−2 and so for |x| sufficiently large there is
a constant κ > 0 such that
∣∣ d
dx
( R(x)
V ′(x)
)∣∣ ≤ κ|x|−2. Thus∣∣∣∣eV (z)
∫ (+∞)·e2πi k
D
z
R(x)e−V (x)dx
∣∣∣∣ ≤ |R(z)|
|V ′(z)|
+ κ
∫ (+∞)·e2πi k
D
z
|x|−2|dx| −→ 0
as z → ∞. This limit holds so long as z is confined to the sector 2π
k− 1
2
D ≤ argφ(z) ≤ 2π
k+ 1
2
D ,
however since there are only finitely many such sectors this limit thus holds uniformly in argφ(z)
(and so uniformly in arg z). Thus R ≡ 0.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 9
Finally, this shows that eV (z)
∫ +∞
z Q(x)e−V (x)dx is a polynomial, since if we factorise Q and
integrate by parts, we find
eV (z)
∫ +∞
z
Q(x)e−V (x)dx = F1(z) + · · ·+ Fℓ(z),
which is a polynomial of degree degQ−D + 1. ■
Lemma 2.5. Let Ṽ (w) = 1
γV (w) = wD + O
(
wD−1
)
and let φ(w) := Ṽ (w)
1
D . There exists
a ϱ > 0 sufficiently large such that φ is analytic and injective on
Uϱ := {z ∈ C : |z| > ϱ}
and such that the following holds. For each z ∈ Uϱ such that
2π(k− 1
2
)
D ≤ argφ(z) ≤ 2π(k+ 1
2
)
D , there
is a of contour τz starting at z and tending to (+∞) · e2πi
k
D such that Re(V (w)) is non-decreasing
along the contour and
∫
τz
|x|−2|dx| → 0 as |z| → ∞ uniformly in arg z.
Proof. Let
τ̃(s) =
{
|φ(z)| exp
(
i(1− s) argφ(z) + is2πkD
)
, s ∈ [0, 1],
(|φ(z)|+ s− 1) exp
(
i2πkD
)
, s ∈ [1,+∞).
Then we choose our contour to be the preimage of τ̃ , that is τ = φ−1 ◦ τ̃ . From this, we see that
V (τ(s)) = γτ̃(s)D. A short calculation then shows that for s ∈ (0, 1)
d
ds
Re(V (τ(s))) = γ|φ(z)|DD
(
argφ(z)− 2πk
D
)
sin
(
D(1− s)
(
argφ(z)− 2πk
D
))
≥ 0.
It is also easy to show for s ∈ (1,+∞), d
ds
(
Re τ̃(s)D
)
≥ 0. Furthermore, φ(w) = w(1 + o(1))
and φ′(w) = 1 + o(1) uniformly on Uϱ as ϱ→ +∞, and so
∫
τ |x|
−2|dx| → 0 as |z| → ∞ uniformly
in arg z. ■
A central result in the theory of orthogonal polynomials is that their zeros are simple and
lie in the support of the weight function. The following proposition gives a partial extension of
this to dual skew-orthogonal polynomials.
Proposition 2.6. Ψ2n has at least 2n + 1 real zeros (not counting multiplicity), of which at
least 2n+2− D
2 must be simple. Ψ2n+1 has at least 2n real zeros (not counting multiplicity), of
which at least 2n− D
2 must be simple.
Proof. Let x1, . . . , xℓ be the collection of real zeros of Ψ2n with odd multiplicity, i.e., exactly
those points on R at which Ψ2n changes sign, where we do not repeat according to multiplicity.
Then Ψ2n(x)(x−x1) · · · (x−xℓ) has the same sign everywhere and vanishes only on a finite set,
hence∫
R
Ψ2n(x)(x− x1) · · · (x− xℓ)e
−2V (x)dx ̸= 0.
However, if ℓ ≤ 2n then the above integral vanishes, thus ℓ ≥ 2n + 1. Furthermore, it is easily
seen that at least 2n+2− D
2 of these must be simple. A similar argument works for Ψ2n+1. ■
We now arrive at a pair of Riemann–Hilbert problems, for the even and odd dual skew-
orthogonal polynomials, respectively. In what follows, it is more convenient to ensure the
uniqueness of the sequence by fixing the next-to-leading coefficient of Ψ2n+1 to be zero rather
than P2n+1. We begin with the “even problem”, so called because the (1, 1) matrix element of
the solution will turn out to be Ψ2n.
10 A. Little
Riemann–Hilbert Problem 2.7 (even problem). Let V be a real polynomial of even de-
gree D ≥ 2 and positive leading coefficient, Γj be defined by (2.1) and (2.2), and
Γ =
D−1⋃
j=1
Γj (2.4)
with the orientations of Γj inherited by Γ (see, e.g., Figure 1). We look for a (D+1)× (D+1)
matrix valued function
An : C \ Γ −→ MD+1(C)
such that
(1) An is analytic on C \ Γ (that is, its matrix elements are analytic on C \ Γ).
(2) An has continuous boundary values up to Γ\{0}. That is, given x ∈ Γ\{0}, as C\Γ ∋ z → x
from either the + (left) side or − (right) side non-tangentially, the limit of An(z) exists.
These limits are denoted
A+
n : Γ \ {0} −→ MD+1(C), A−
n : Γ \ {0} −→ MD+1(C)
and are continuous functions. “Left” and “right” mean relative to the orientation of the
contour. The boundary values are related by the “jump matrix”
A+
n (x) = A−
n (x)
1 e−2V χR e−V χΓ1 . . . e−V χΓD−1
1
1
. . .
1
, x ∈ Γ \ {0}.
(3) An is bounded in a neighbourhood of 0.
(4) We have the asymptotic normalisation
An(z) =
(
I+O
(
z−1
))
z2n+D−1
z−2n
z−1
. . .
z−1
, z → ∞.
Lemma 2.8. If a solution to Riemann–Hilbert Problem 2.7 exists, then it is unique.
Proof. This method is standard but we repeat it for completeness. We note that the jump ma-
trix has determinant 1 so detAn(z) has no jump across Γ \ {0} and takes continuous boundary
values, so by Morera’s theorem is analytic everywhere except possibly at 0. However, the re-
quirement of boundedness in a neighbourhood of 0 implies that the singularity at 0 is removable.
Hence z 7→ detAn(z) is entire. Furthermore, we have the normalisation detAn(z) = 1+O
(
z−1
)
,
as z → ∞ and so detAn(z) ≡ 1. Then let Ãn be another solution. Then the function
z 7→ Ãn(z)A
−1
n (z) has no jump across Γ \ {0} and obtains continuous boundary values, so by
Morera’s theorem is analytic everywhere except possibly at 0. Again by the requirement of
boundedness of Ãn and An in a neighbourhood of 0 and that detAn(z) ≡ 1, we see that the
singularity at 0 is removable. Hence z 7→ Ãn(z)A
−1
n (z) is actually entire. Finally, since
Ãn(z)A
−1
n (z) = I+O
(
z−1
)
,
we find that Ãn(z)A
−1
n (z) ≡ I. ■
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 11
The jump condition implies that (An)11 is an entire function which scales like z2n+D−1 as
z → ∞. It is therefore a monic polynomial of degree 2n+D − 1 by Liouville’s theorem,
(An)11 := Q ∈ PC
2n+D−1.
The Plemelj formula then implies that the first row of An is(
Q,CR
(
e−2VQ
)
, CΓ1
(
e−VQ
)
, . . . , CΓD−1
(
e−VQ
))
.
From the asymptotic condition
CΓj
(
e−VQ
)
(z) = O
(
z−2
)
, ∀j ∈ [[1, D − 1]],
we see, by Proposition 2.4, that there exists a unique P ∈ PC of degree exactly 2n such
that Q = eV d
dx
(
P e−V
)
. Finally, the normalisation
CR
(
e−2VQ
)
(z) = O
(
z−2n−1
)
, z → ∞
implies that ⟨X k, P ⟩4 = 0 for k ∈ [[0, , 2n− 1]], and so P is skew-orthogonal. Thus Q = Ψ2n.
For the second row, we see that (An)21 := Q̃ is a polynomial of degree at most 2n +D − 2.
Following the same procedure as above, we see that the second row is(
Q̃, CR
(
e−2V Q̃
)
, CΓ1
(
e−V Q̃
)
, . . . , CΓD−1
(
e−V Q̃
))
,
where now Q̃ = −2πiγD
hn−2
Ψ2n−2. For the remaining rows, we see that (An)j+2,1 := R
(j)
n ∈ PC
2n+D−2
must be a polynomial of degree at most 2n+D − 2. To show there exists a unique solution to
Riemann–Hilbert Problem 2.7, let us introduce the following linear functionals:
αj : PC
2n+D−2 −→ C, βj : PC
2n+D−2 −→ C,
where
αj(P )
def
=
∫
R
e−2V (x)xjP (x)dx, j ∈ [[0, 2n− 1]], (2.5)
βj(P )
def
= − 1
2πi
∫
Γj
e−V (x)P (x)dx, j ∈ [[1, D − 1]]. (2.6)
Lemma 2.9. The linear functionals α0, . . . , α2n−1, β1, . . . , βD−1 form a linearly independent set(
and thus span
(
PC
2n+D−2
)∗)
.
Proof. To show this, we need only show that
2n−1⋂
j=0
kerαj ∩
D−1⋂
k=1
kerβk = 0.
Thus let Q ∈ PC
2n+D−2 be such that α0(Q) = · · · = α2n−1(Q) = β1(Q) = · · · = βD−1(Q) = 0.
Thus by Proposition 2.4, Q = eV d
dx
(
P e−V
)
for some P ∈ PC
2n−1. Furthermore,
〈
X 0, P
〉
4
= · · · =〈
X 2n−1, P
〉
4
= 0. Hence P is skew-orthogonal to all of PC
2n−1. Thus by the non-degeneracy of
the inner product ⟨·, ·⟩4, P ≡ 0, and hence Q ≡ 0. ■
Let R
(j)
n ∈ PC
2n+D−2 be the unique polynomial such that α0
(
R
(j)
n
)
= · · · = α2n−1
(
R
(j)
n
)
= 0
and
βk
(
R(j)
n
)
= δkj , ∀k ∈ [[1, D − 1]].
That such a polynomial exists and is unique follows from the linear independence of these
functionals.
12 A. Little
Theorem 2.10. The unique solution of the even Riemann–Hilbert Problem 2.7 is
An =
Ψ2n CR
(
e−2VΨ2n
)
CΓ1
(
e−VΨ2n
)
· · · CΓD−1
(
e−VΨ2n
)
−2πiγD
hn−1
Ψ2n−2 −2πiγD
hn−1
CR
(
e−2VΨ2n−2
)
−2πiγD
hn−1
CΓ1
(
e−VΨ2n−2
)
· · · −2πiγD
hn−1
CΓD−1
(
e−VΨ2n−2
)
R
(1)
n CR
(
e−2VR
(1)
n
)
CΓ1
(
e−VR
(1)
n
)
· · · CΓD−1
(
e−VR
(1)
n
)
...
R
(D−1)
n CR
(
e−2VR
(D−1)
n
)
CΓ1
(
e−VR
(D−1)
n
)
· · · CΓD−1
(
e−VR
(D−1)
n
)
.
Lemma 2.11 (symmetry for even case). Let P ∈ MD−1(R) be the permutation matrix on RD−1
for which Pek = eD−k for k = 1, . . . , D − 1. Here ek = (0, . . . , 0, 1, 0, . . . , 0)T, where the 1 lies
in the k-th entry. Let
Q := diag(−1, 1)⊕ P =
−1 0
0 1
P
∈ MD+1(R).
Note that Q = QT = Q−1 is an orthogonal matrix. Then we have the relation An(z) = QAn(z)Q.
Proof. Let Ãn(z) = QAn(z)Q. We show that Ãn solves the even Riemann–Hilbert problem
and thus by uniqueness (see Lemma 2.8) Ãn = An. Clearly, Ãn is analytic on C \ Γ and it is
easy to see that
Ãn(z) =
(
I+O
(
z−1
))
z2n+D−1
z−2n
z−1
. . .
z−1
, z → ∞.
Furthermore, Ãn is clearly bounded at 0 since An is.
Because V is a polynomial with real coefficients, V (x) = V (x). Furthermore, χR(x) = χR(x)
and χΓk
(x) = χΓD−k
(x). Now for x ∈ Γ \ {0}, we have(
Ãn
)
+
(x) = Q(An)−(x)Q
= Q(An)+(x)
1 −e−2V χR −e−V χΓD−1
. . . −e−V χΓ1
1
1
. . .
1
Q
=
(
Ãn
)
−(x)
1 e−2V χR e−V χΓ1 . . . e−V χΓD−1
1
1
. . .
1
.
Thus Ãn = An. ■
Remark 2.12. Our method assumes V is a polynomial, however there are interesting cases,
e.g., Jacobi weights, that do not fit into this class. How might our analysis be extended? The
natural way would be to redefine Ψn and R
(j)
n so that these are no longer polynomials. For
example, if V ′ was rational with simple poles at λ1, . . . , λk, then Ψn would also be rational with
simple poles, with the property that Resz=λi
Ψn(z)e
−V (z) = 0 for all i = 1, . . . , k and similarly
for the auxiliary functions R
(j)
n . Investigation of this is left as a topic for future research.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 13
We now move onto the “odd problem”, so called because it will later turn out that the (1, 1)
matrix element of the solution is Ψ2n+1.
Riemann–Hilbert Problem 2.13 (odd problem). Let V be a real polynomial of even de-
gree D ≥ 2 and positive leading coefficient, and let
Σ = T ∪
D−1⋃
j=1
Γj ,
where Γj have the specified orientations and T is positively oriented (see, e.g., Figure 2). Let
ω = e
2πi
D be a primitive root of unity and let
S =
{
0, 1, ω, ω2, . . . , ωD−1
}
be the points of self-intersection of Σ. We look for a (D + 2)× (D + 2) matrix valued function
Bn : C \ Σ −→ MD+2(C)
such that
(1) Bn is analytic on C \ Σ (that is, its matrix elements are analytic on C \ Σ).
(2) Bn has continuous boundary values up to Σ\S. That is, given x ∈ Σ\S, as C\Σ ∋ z → x
from either the + (left) side or − (right) side non-tangentially, the limit of Bn(z) exists.
These limits are denoted
B+
n : Σ \ S −→ MD+2(C), B−
n : Σ \ S −→ MD+2(C)
and are continuous functions. “Left” and “right” mean relative to the orientation of the
contour. The boundary values are related by the “jump matrix”
B+
n (x) = B−
n (x)
1 e−2V χR e−V χΓ1 . . . e−V χΓD−1
X−2n−DχT
1
1
. . .
1
1
,
x ∈ Σ \ S.
(3) Bn is bounded in a neighbourhood of S.
(4) We have the asymptotic normalisation
Bn(z) =
(
I+O
(
z−1
))
z2n+D
z−2n
z−1
. . .
z−1
, z → ∞.
The method of solution of this Riemann–Hilbert problem is very similar to the even case.
The asymptotic condition implies that (Bn)11 = Q is a monic polynomial of degree 2n + D.
From the jump condition and the Plemelj formula we find that the first row is(
Q,CR
(
e−2VQ
)
, CΓ1
(
e−VQ
)
, . . . , CΓD−1
(
e−VQ
)
, CT
(
X−2n−DQ
))
.
From the normalisation CΓj
(
e−VQ
)
(z) = O
(
z−2
)
for all j ∈ [[1, D− 1]], we find that there exists
a polynomial P of degree 2n+ 1 such that Q = eV d
dx
(
P e−V
)
.
14 A. Little
C0
C2
C1
C4
C3
C5
C6
C7
T
Figure 2. The contours of the odd Riemann–Hilbert problem, here shown for the case D = 8. Orienta-
tions of the contours are indicated by arrows.
Remark 2.14. The scaling CT
(
X−2n−DQ
)
(z) = O
(
z−2
)
implies that
Q(x) = x2n+D +O
(
x2n+D−2
)
, z → ∞.
This fixes the next-to-leading coefficient of P
(
where Q = eV d
dx
(
P e−V
))
though not necessarily
to 0. This explains the remark made in the introduction that fixing the next-to-leading coefficient
of P to 0 is not always the most convenient choice. The addition of the contour T could also be
done for the Riemann–Hilbert problem of Pierce [38] in order to guarantee a unique solution.
Finally, the scaling CR
(
e−2VQ
)
(z)=O
(
z−2n−1
)
implies that
〈
X 0, P
〉
4
= · · ·=
〈
X 2n−1, P
〉
4
=0,
so P is indeed skew-orthogonal. Thus Q = Ψ2n+1.
In the same manner as for the even case, we find that the second row is(
Q̃, CR
(
e−2V Q̃
)
, CΓ1
(
e−V Q̃
)
, . . . , CΓD−1
(
e−V Q̃
)
, CT
(
X−2n−DQ̃
))
,
where Q̃ = −2πiγD
hn−1
Ψ2n−2. Indeed, the remaining rows except the final one are the same as
in the even case. To show a unique solution in all remaining rows, let us introduce the linear
functional βD : PC
2n+D−2 −→ C defined by
βD(P )
def
= − 1
2πi
∮
T
P (z)z−2n−Ddz.
Now let α0, . . . , α2n−1 and β1, . . . , βD−1 act on PC
2n+D−1 according to the formulas given in (2.5)
and (2.6). That the 2nd to (D+2)-th rows have a unique solution follows from the lemma below.
Lemma 2.15. Consider the linear functionals α0, . . . , α2n−1 and β1, . . . , βD viewed as acting
on PC
2n+D−1 according to (2.5) and (2.6). Then
2n−1⋂
j=0
kerαj ∩
D⋂
k=1
kerβk = 0.
Proof. If βD(Q) = 0, then Q ∈ PC
2n+D−2. The proof then reduces to that of Lemma 2.9. ■
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 15
Theorem 2.16. The unique solution of the odd Riemann–Hilbert Problem 2.13 is then
Bn =
Ψ2n+1 CR
(
e−2VΨ2n+1
)
CΓ1
(
e−VΨ2n+1
)
· · · CT
(
X−2n−DΨ2n+1
)
−2πiγD
hn−1
Ψ2n−2 −2πiγD
hn−1
CR
(
e−2VΨ2n−2
)
−2πiγD
hn−1
CΓ1
(
e−VΨ2n−2
)
· · · −2πiγD
hn−1
CT
(
X−2n−DΨ2n−2
)
R
(1)
n CR
(
e−2VR
(1)
n
)
CΓ1
(
e−VR
(1)
n
)
· · · CT
(
X−2n−DR
(1)
n
)
...
R
(D−1)
n CR
(
e−2VR
(D−1)
n
)
CΓ1
(
e−VR
(D−1)
n
)
· · · CT
(
X−2n−DR
(D−1)
n
)
R
(D)
n CR
(
e−2VR
(D)
n
)
CΓ1
(
e−VR
(D)
n
)
· · · CT
(
X−2n−DR
(D)
n
)
.
Note that these polynomials R
(1)
n , . . . , R
(D−1)
n are the same polynomials that appear in the
even problem, so we have not made a confusion of notation. Note also that the integrals in the
final column can be computed explicitly.
Proposition 2.17. R
(D)
n = −Ψ2n.
Proof. From the normalisation, we see that R
(D)
n is a polynomial of degree at most 2n+D−1,
and from CT
(
X−2n−DR
(D)
n
)
(z) = z−1 +O
(
z−2
)
, we find that R
(D)
n (x) = −Q(x), where Q is
a monic polynomial of degree 2n+D − 1. From the fact that
β1
(
R(D)
n
)
= · · · = βD−1
(
R(D)
n
)
= 0,
we find that Q = eV d
dx
(
P e−V
)
for some polynomial P of degree exactly 2n. Finally, the require-
ment that
α0
(
R(D)
n
)
= · · · = α2n−1
(
R(D)
n
)
= 0
implies that P is skew-orthogonal to all of PC
2n−1. ■
From this, one might think the Bn problem is strictly superior to the An problem, since the
former encodes both Ψ2n+1 and Ψ2n, however our Christoffel–Darboux type formula (3.1) implies
that from the point of view of random matrix theory it is completely sufficient to study An.
Lemma 2.18. 2n ≤ degR
(j)
n ≤ 2n+D − 2 for j ∈ [[1, D − 1]].
Proof. Suppose by way of contradiction that degR
(j)
n ≤ 2n − 1. Then by the conditions
α0
(
R
(j)
n
)
= · · · = α2n−1
(
R
(j)
n
)
= 0, we find that R
(j)
n ≡ 0. But this contradicts βj
(
R
(j)
n
)
= 1. ■
Remark 2.19. For the case n = 0, the polynomials R
(1)
0 , . . . , R
(D−1)
0 can be constructed as
follows. Let M be the matrix
M = (Mij)
D−1
i,j=1 =
(∫
Γi
xj−1e−V (x)dx
)D−1
i,j=1
,
then we have
R
(j)
0 (x) = −2πi
D−1∑
k=1
(
M−1
)
kj
xk−1, j ∈ [[1, D − 1]].
Lemma 2.20 (symmetry for odd problem). Let P ∈ MD−1(R) be defined as in Lemma 2.11
and let
Q := diag(−1, 1)⊕ P ⊕ (−1) =
−1 0
0 1
P
−1
∈ MD+2(R).
Note that Q = QT = Q−1 is an orthogonal matrix. Then we have the relation Bn(z) = QBn(z)Q.
16 A. Little
Proof. Morally, the same as for the even case. ■
Finally, our results allow us to characterise the “dual” skew-orthogonal polynomials Ψ2n and
auxiliary polynomials R
(j)
n in terms of the corresponding sequence of orthogonal polynomials.
This is morally the same construction as in Remark 2.19.
Corollary 2.21 (skew-orthogonal polynomials in terms of orthogonal polynomials). Let {Hn}n∈N
be the sequence of monic orthogonal polynomials corresponding to the weight e−2V , where V (x) =
γxD +O
(
xD−1
)
is a real polynomial of even degree D ≥ 2 and γ > 0. That is,
Hn(x) = xn +O
(
xn−1
)
and ∫
R
Hn(x)Hm(x)e−2V (x)dx = 0, ∀n,m ∈ N, n ̸= m.
Let M be the matrix
M = (Mjk)
D−1
j,k=1 =
(∫
Γj
H2n+k−1(z)e
−V (z)dz
)D−1
j,k=1
,
and let λn be the next-to-leading coefficient of Hn+D, i.e.,
Hn+D(x) = xn+D + λnx
n+D−1 +O
(
xn+D−2
)
.
Then the corresponding “dual” skew-orthogonal polynomials and auxiliary polynomials may be
expressed
Ψ2n = H2n+D−1 −
D−1∑
k,j=1
[(
M−1
)
kj
∫
Γj
H2n+D−1(z)e
−V (z)dz
]
H2n+k−1,
Ψ2n+1 = H2n+D − λnH2n+D−1
−
D−1∑
k,j=1
[(
M−1
)
kj
∫
Γj
(H2n+D(z)− λnH2n+D−1(z))e
−V (z)dz
]
H2n+k−1,
R(j)
n = −2πi
D−1∑
k=1
(
M−1
)
kj
H2n+k−1.
Indeed, these {Hk}k∈N are exactly the “natural basis” identified by Adler and van Moer-
beke [3]. A similar construction to Corollary 2.21 in which orthogonal and skew-orthogonal
polynomials were related was considered recently in the planar case by [4], however we note that
in Corollary 2.21 the relation is really between orthogonal polynomials and dual skew-orthogonal
polynomials.
Example 2.22. In the case D = 2, for the potential V (z) = 1
2z
2, we can write the corresponding
orthogonal polynomials explicitly as
Hn(z) =
(−1)n
2n
ez
2 dn
dzn
e−z2 .
Then using
∫
RH2n(x)e
− 1
2
x2
dx =
√
2
(
n− 1
2
)
!, one finds that
Ψ2n = H2n+1, Ψ2n+1 = H2n+2 −
(
n+
1
2
)
H2n, P2n = n!
n∑
k=0
1
k!
H2k,
R(1)
n = − 2πi√
2
(
n− 1
2
)
!
H2n, P2n+1 = H2n+1.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 17
These formulas for P2n and P2n+1 already appeared in the paper of Adler, Forrester, Nagao and
van Moerbeke [1].
What we have done in the preceding discussion amounts to representing Ψ2n and Ψ2n+1 as
“multiple-orthogonal polynomials of Type II” (see, e.g., [34] for a definition of this term). Let
us summarise this with the following corollary.
Corollary 2.23 (dual skew-orthogonal polynomials as Type II multiple-orthogonal polynomi-
als). Consider the following problem. Find a monic polynomial Q of degree exactly 2n+D − 1
such that∫
R
xkQ(x)e−2V (x)dx = 0, ∀k ∈ [[0, 2n− 1]],∫
Γj
Q(x)e−V (x)dx = 0, ∀j ∈ [[1, D − 1]].
Then this problem is uniquely solved by Q = Ψ2n. Similarly, if we are asked to find a monic
polynomial Q of degree exactly 2n+D such that∫
R
xkQ(x)e−2V (x)dx = 0, ∀k ∈ [[0, 2n− 1]],∫
Γj
Q(x)e−V (x)dx = 0, ∀j ∈ [[1, D − 1]],∫
T
Q(x)x−2n−Ddx = 0,
then this problem is uniquely solved by Q = Ψ2n+1. We should note that our case differs from
usual multiple-orthogonality because our “measures” are not all supported on R.
Following the standard theory of multiple-orthogonal polynomials [43], it is natural to con-
sider the Riemann–Hilbert problems satisfied by Ân := A−T
n and B̂n := B−T
n (inverse transpose).
Riemann–Hilbert Problem 2.24 (dual even problem). By virtue of the fact that detAn ≡ 1,
Ân
def
= A−T
n satisfies the following Riemann–Hilbert problem:
(1) Ân is analytic on C \ Γ (that is, its matrix elements are analytic on C \ Γ).
(2) Ân has continuous boundary values up to Γ\{0}. That is, given x ∈ Γ\{0}, as C\Γ ∋ z → x
from either the + (left) side or − (right) side non-tangentially, the limit of Ân(z) exists.
These limits are denoted
Ân
+
: Γ \ {0} −→ MD+1(C), Ân
−
: Γ \ {0} −→ MD+1(C)
and are continuous functions. “Left” and “right” mean relative to the orientation of the
contour. The boundary values are related by the “jump matrix”,
Ân
+
(x) = Ân
−
(x)
1
−e−2V χR 1
−e−V χΓ1 1
...
. . .
−e−V χΓD−1
1
, x ∈ Γ \ {0}.
(3) Ân is bounded in a neighbourhood of 0.
18 A. Little
(4) We have the asymptotic normalisation
Ân(z) =
(
I+O
(
z−1
))
z−2n−D+1
z2n
z
. . .
z
, z → ∞.
In what follows we shall only be interested in computing the second column of Ân (the
motivation for this is that this column will appear in the upcoming Christoffel–Darboux-type
formula), though it is an easy exercise for the reader to compute the remaining columns. By
virtue of the jump condition, the 2nd to (D + 1)-th columns all have no jump across Γ \ {0}
and by boundedness around 0 we see that the singularity at 0 is removable. Hence the 2nd
to (D + 1)-th columns are entire functions, indeed polynomials by the asymptotic condition,(
Ân
)
i2
(z) = z2nδi,2 +O
(
z2n−1
)
, i ∈ [[1, D + 1]],(
Ân
)
ij
(z) = zδij + cij , i ∈ [[1, D + 1]] and j ∈ [[3, D + 1]],
where the cij are constants. The jump condition on the first column may be solved by Plemelj’s
formula
(
Ân
)
i1
= −CR
((
Ân
)
i2
e−2V
)
−
D+1∑
j=3
cijCΓj−2
(
e−V
)
− χi≥3CΓi−2
(
X e−V
)
, i ∈ [[1, D + 1]].
The asymptotic condition
(
Ân
)
11
(z) = z−2n−D+1
(
1 +O
(
z−1
))
yields the conditions
∫
R
(
Ân
)
12
(x)xke−2V (x)dx+
D+1∑
j=3
c1j
∫
Γj−2
xke−V (x)dx = 0, ∀k ∈ [[0, 2n+D − 3]].
By taking linear combinations of these equations, we can replace xk with any polynomial of
degree ≤ 2n+D−3; in particular, we can replace it with Ψ0, . . . ,Ψ2n−2. This implies that
(
Ân
)
12
is a scalar multiple of P2n−2. This multiple is fixed by∫
R
(
Ân
)
12
(x)Ψ2n−1(x)e
−2V (x)dx = 2πi,
and so(
Ân
)
12
= −2πiγD
hn−1
P2n−2. (2.7)
By a similar argument, we see that(
Ân
)
22
= P2n. (2.8)
The remaining entries in the second column are polynomials of degree at most 2n − 1. These
satisfy the conditions∫
R
(
Ân
)
i2
(x)Ψk(x)e
−2V (x)dx+
∫
Γi−2
xΨk(x)e
−V (x)dx = 0,
∀k ∈ [[0, 2n− 1]] and i ≥ 3.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 19
Expanding this in the basis P0, . . . , P2n−1, we find(
Ân
)
i2
= γD
n−1∑
k=0
h−1
k
(
P2k
∫
Γi−2
xΨ2k+1(x)e
−V (x)dx− P2k+1
∫
Γi−2
xΨ2k(x)e
−V (x)dx
)
i ∈ [[3, D + 1]]. (2.9)
What the above calculation demonstrates is that P2n may be thought of as a Type I multiple-
orthogonal polynomial (again see [34] for a definition of this term). Let us formalise this with
the following corollary.
Corollary 2.25 (even skew-orthogonal polynomials as Type I multiple-orthogonal polynomials).
Consider the following problem. Find a monic polynomial P of degree exactly 2n and a collection
of constants c1, . . . , cD−1 such that∫
R
xkP (x)e−2V (x)dx+
D−1∑
j=1
cj
∫
Γj
xke−V (x)dx = 0, ∀k ∈ [[0, 2n+D − 2]].
Then this problem has a unique solution (P, c1, . . . , cD−1), where P = P2n.
It is curious that Types I and II multiple orthogonality both make their appearance here
(a similar situation, in which both Types I and II multiple orthogonality appear in the study
of the same ensemble, was observed recently in certain planar ensembles [5, 33]). On the other
hand, it is not clear if it is possible to think of P2n+1 as a Type I multiple-orthogonal polynomial
and, as we shall soon see, P2n+1 (or any multiple of it) does not appear as a matrix element
of B−1
n . It is possible to characterise P2n+1 by a combination of both Types I and II orthogonality
conditions, as follows. We look for a monic polynomial P of degree 2n + 1 and a collection of
constants c1, . . . , cD−1 such that∫
R
P (x)xke−2V (x)dx+
D−1∑
j=1
cj
∫
Γj
xke−V (x)dx = 0, ∀k ∈ [[0, 2n+D − 2]]
and ∫
T
P (x)e−V (x) d
dx
(
x−2n−DeV (x)
)
dx = 0.
(P, c1, . . . , cD−1) = (P2n+1, ∗, . . . , ∗) is the unique solution of this problem. However, this con-
struction seems somewhat ugly and we shall not investigate the implications of this here.
Riemann–Hilbert Problem 2.26 (dual odd problem). By virtue of the fact that detBn ≡ 1,
B̂n
def
= B−T
n satisfies the following Riemann–Hilbert problem:
(1) B̂n is analytic on C \ Σ (that is, its matrix elements are analytic on C \ Σ).
(2) B̂n has continuous boundary values up to Σ\S. That is, given x ∈ Σ\S, as C\Σ ∋ z → x
from either the + (left) side or − (right) side non-tangentially, the limit of B̂n(z) exists.
These limits are denoted
B̂n
+
: Σ \ S −→ MD+2(C), B̂n
−
: Σ \ S −→ MD+2(C)
and are continuous functions. “Left” and “right” mean relative to the orientation of the
contour. The boundary values are related by the “jump matrix”
B̂n
+
(x) = B̂n
−
(x)
1
−e−2V χR 1
−e−V χΓ1 1
...
. . .
−e−V χΓD−1
1
−X−2n−DχT 1
, x ∈ Σ \ S.
20 A. Little
(3) B̂n is bounded in a neighbourhood of S.
(4) We have the asymptotic normalisation
B̂n(z) =
(
I+O
(
z−1
))
z−2n−D
z2n
z
. . .
z
, z → ∞.
Here again we restrict ourselves to computing the second column. We shall simply state the
result since the calculation is practically the same as in the case of Ân,(
B̂n
)
12
= 0,
(
B̂n
)
22
= P2n,
(
B̂n
)
i2
=
(
Ân
)
i2
, i ∈ [[3, D + 1]],(
B̂n
)
D+2,2
=
2πiγD
hn−1
P2n−2.
3 Christoffel–Darboux type formula
In this section, we prove the following theorem.
Theorem 3.1 (Christoffel–Darboux type formula). Let the pre-kernel Sn be defined by (1.4).
Then
Sn(x, y) = − 1
4πi
e−V (x)−V (y)
(
A−1
n (x)An(y)
)
21
x− y
, (3.1)
where An is the unique solution of Riemann–Hilbert Problem 2.7. Note that we do not specify
whether we take the + or − boundary value of
(
A−1
n (x)An(y)
)
21
since this will turn out to be an
entire function.
Remark 3.2. We call this a “Christoffel–Darboux type” formula because an analogous for-
mula appears in the theory of orthogonal polynomials (β = 2), i.e., if one takes An to be the
2× 2 Fokas–Its–Kitaev Riemann–Hilbert problem [27], then (3.1) yields the classical Christoffel–
Darboux formula, up to a factor of 1
2 .
Remark 3.3. We could, if we like, have expressed the Christoffel–Darboux type formula (3.1) in
terms of the solution of the odd Riemann–Hilbert Problem 2.13, since we see from our solution
to Bn and B̂n that
(
A−1
n (x)An(y)
)
21
=
(
B−1
n (x)Bn(y)
)
21
.
To prove this result, we need to derive a variety of relations. First note that the polynomial
R
(j)
k −R
(j)
n for k = 0, . . . , n− 1 is in the kernel of βj for all j and has degree at most 2n+D− 2.
It is therefore in the span of Ψ0, . . . ,Ψ2n−1. Thus write
R
(j)
k −R(j)
n =
n−1∑
i=0
η
(j)
ik Ψ2i +
n−1∑
i=0
ξ
(j)
ik Ψ2i+1.
We then have
η
(j)
ik =
γD
hi
∫
R
R
(j)
k P2i+1e
−2V dx, ξ
(j)
ik = −γD
hi
∫
R
R
(j)
k P2ie
−2V dx.
Note that for i ≤ k − 1, we have η
(j)
ik = ξ
(j)
ik = 0. We may thus write
R
(j)
k = R(j)
n +
n−1∑
i=k
η
(j)
ik Ψ2i +
n−1∑
i=k
ξ
(j)
ik Ψ2i+1.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 21
Next consider XΨ2k+1−Ψ2k+2. This has degree at most 2n+D and is orthogonal to PC
2k−2.
Then XΨ2k+1−Ψ2k+2−
∑D−1
j=1 βj(XΨ2k+1)R
(j)
k has degree at most 2n+D and is in the kernel
of βj for all j = 1, . . . , D− 1. It is thus a linear combination of Ψ2k−2, Ψ2k and Ψ2k+1. We thus
find
XΨ2k+1 = Ψ2k+2 +
D−1∑
j=1
βj(XΨ2k+1)R
(j)
k + akΨ2k+1 + bkΨ2k + ckΨ2k−2
= Ψ2k+2 + akΨ2k+1 + bkΨ2k + ckΨ2k−2 +
D−1∑
j=1
βj(XΨ2k+1)R
(j)
n
+
D−1∑
j=1
βj(XΨ2k+1)
n−1∑
i=k
η
(j)
ik Ψ2i +
D−1∑
j=1
βj(XΨ2k+1)
n−1∑
i=k
ξ
(j)
ik Ψ2i+1.
These coefficients ak, bk and ck are unique as can be seen from the formulas
ak = −γD
hk
∫
R
xP2kΨ2k+1e
−2V dx−
D−1∑
j=1
βj(XΨ2k+1)ξ
(j)
kk ,
bk =
γD
hk
∫
R
xΨ2k+1P2k+1e
−2V dx−
D−1∑
j=1
βj(XΨ2k+1)η
(j)
kk , ck = − hk
hk−1
.
Similarly, XΨ2k − Ψ2k+1 has degree at most 2n + D − 1 and is orthogonal to PC
2k−1. Thus
XΨ2k −Ψ2k+1 −
∑D−1
j=1 βj(XΨ2k)R
(j)
k is in the kernel of all βj and is orthogonal to PC
2k−1.
Thus it must be proportional to Ψ2k. This gives
XΨ2k = Ψ2k+1 +
D−1∑
j=1
βj(XΨ2k)R
(j)
k + ãkΨ2k
= Ψ2k+1 + ãkΨ2k +
D−1∑
j=1
βj(XΨ2k)R
(j)
n +
D−1∑
j=1
βj(XΨ2k)
n−1∑
i=k
η
(j)
ik Ψ2i
+
D−1∑
j=1
βj(XΨ2k)
n−1∑
i=k
ξ
(j)
ik Ψ2i+1. (3.2)
As before, ãk is unique and may be written
ãk =
γD
hk
∫
R
xΨ2kP2k+1e
−2V dx−
D−1∑
j=1
βj(XΨ2k)η
(j)
kk .
Equation (3.2) also implies the intriguing identity
2 = −
D−1∑
j=1
βj(XΨ2k)ξ
(j)
kk . (3.3)
These formulas imply “dual” formulas for P2k and P2k+1. Let us begin with the even case. XP2k
is monic of degree 2k + 1. Thus consider XP2k − P2k+1 which has degree at most 2k. Thus we
may expand
XP2k − P2k+1 =
k∑
m=0
µmkP2m +
k−1∑
m=0
λmkP2m+1.
22 A. Little
We then find
µmk = −γD
hm
∫
R
xP2kΨ2m+1e
−2V dx =
hk
hm
D−1∑
j=1
βj(XΨ2m+1)ξ
(j)
km + akδkm,
λmk =
γD
hm
∫
R
xP2kΨ2me−2V dx = − hk
hm
D−1∑
j=1
βj(XΨ2m)ξ
(j)
km.
Finally, consider XP2k+1 − P2k+2. This has degree at most 2k + 1 and so may be expanded as
follows:
XP2k+1 − P2k+2 =
k∑
m=0
µ̃mkP2m +
k∑
m=0
λ̃mkP2m+1.
Then after a calculation, we find
µ̃mk = δk,m+1ck −
hk
hm
D−1∑
j=1
βj(XΨ2m+1)η
(j)
km − bkδkm,
λ̃mk =
hk
hm
D−1∑
j=1
βj(XΨ2m)η
(j)
km + ãkδkm.
Let us now derive a Christoffel–Darboux type formula for the pre-kernel (1.4). Note that the
pre-kernel may be written
Sn(x, y) = −1
2
γDe−V (x)−V (y)
n−1∑
k=0
P2k(x)Ψ2k+1(y)− P2k+1(x)Ψ2k(y)
hk
.
Then applying our recursion formulas, we find
(x− y)Sn(x, y) = −1
2
γDe−V (x)−V (y)
n−1∑
k=0
(
1
k
− 2
k
− 3
k
+ 4
k
)
,
where
1
k
= h−1
k
(
P2k+1(x) +
k∑
m=0
µmkP2m(x) +
k−1∑
m=0
λmkP2m+1(x)
)
Ψ2k+1(y),
2
k
= h−1
k P2k(x)
(
n−1∑
i=k
D−1∑
j=1
βj(XΨ2k+1)
[
η
(j)
ik Ψ2i(y) + ξ
(j)
ik Ψ2i+1(y)
]
+Ψ2k+2(y) + akΨ2k+1(y) + bkΨ2k(y) + ckΨ2k−2(y) +
D−1∑
j=1
βj(XΨ2k+1)R
(j)
n (y)
)
,
3
k
= h−1
k
(
P2k+2(x) +
k∑
m=0
µ̃mkP2m(x) +
k∑
m=0
λ̃mkP2m+1(x)
)
Ψ2k(y),
4
k
= h−1
k P2k+1(x)
(
Ψ2k+1(y) + ãkΨ2k(y) +
D−1∑
j=1
βj(XΨ2k)R
(j)
n (y)
+
n−1∑
i=k
D−1∑
j=1
βj(XΨ2k)
[
η
(j)
ik Ψ2i(y) + ξ
(j)
ik Ψ2i+1(y)
])
.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 23
Let us then write
n−1∑
k=0
2
k
= h−1
n−1P2n−2(x)Ψ2n(y) + h−1
n−1P2n(x)Ψ2n−2(y)
+
D−1∑
j=1
n−1∑
k=0
βj(XΨ2k+1)h
−1
k P2k(x)R
(j)
n (y) +
n−1∑
k=0
5
k
,
where
5
k
= h−1
k−1P2k−2(x)Ψ2k(y) + h−1
k P2k(x) (akΨ2k+1(y) + bkΨ2k(y))− h−1
k P2k+2(x)Ψ2k(y)
+
k∑
m=0
h−1
m P2m(x)
D−1∑
j=1
βj(XΨ2m+1)
[
η
(j)
kmΨ2k(y) + ξ
(j)
kmΨ2k+1(y)
]
,
and similarly we may write
n−1∑
k=0
4
k
=
D−1∑
j=1
n−1∑
k=0
h−1
k P2k+1(x)βj(XΨ2k)R
(j)
n (y) +
n−1∑
k=0
6
k
,
where
6
k
= h−1
k P2k+1(x)Ψ2k+1(y) + ãkh
−1
k P2k+1(x)Ψ2k(y)
+
k∑
m=0
h−1
m P2m+1(x)
D−1∑
j=1
βj(XΨ2m)
[
η
(j)
kmΨ2k(y) + ξ
(j)
kmΨ2k+1(y)
]
.
Thus
(x− y)Sn(x, y) =
γD
2
e−V (x)−V (y)
(
h−1
n−1P2n−2(x)Ψ2n(y) + h−1
n−1P2n(x)Ψ2n−2(y)
+
D−1∑
j=1
n−1∑
k=0
h−1
k [βj(XΨ2k+1)P2k(x)− βj(XΨ2k)P2k+1(x)]R
(j)
n (y)
)
+
γD
2
e−V (x)−V (y)
n−1∑
k=0
(
− 1
k
+ 5
k
+ 3
k
− 6
k
)
.
The reader may verify that a miraculous cancellation occurs so that
− 1
k
+ 5
k
+ 3
k
− 6
k
= 0.
Finally, using our solutions of An and Ân,
(x− y)Sn(x, y) =
γD
2
e−V (x)−V (y)
(
h−1
n−1P2n−2(x)Ψ2n(y) + h−1
n−1P2n(x)Ψ2n−2(y)
+
D−1∑
j=1
n−1∑
k=0
h−1
k [βj(XΨ2k+1)P2k(x)− βj(XΨ2k)P2k+1(x)]R
(j)
n (y)
)
= − e−V (x)−V (y)
4πi
(
A−1
n (x)An(y)
)
21
.
24 A. Little
Remark 3.4. We refer to this cancellation as “miraculous”, however one could have predicted
that some sort of cancellation must occur simply from the form of the pre-kernel (1.4). Introduce
the integral operator
Sn : L2(R) −→ L2(R), (Snψ)(x) =
∫
R
Sn(x, y)ψ(y)dy, ψ ∈ L2(R).
It is then easily seen that if {Hk}k∈N is the sequence of orthogonal polynomials with respect
to the weight w = e−2V , then Sn
(
Hke
−V
)
= 1
2Hke
−V for k ≤ 2n − 1 and Sn
(
Hke
−V
)
= 0 for
k ≥ 2n+D − 1. It thus follows that
Sn =
1
2
Kn +
2n+D−2∑
j=2n
1
νj
Sn
(
Hje
−V
)
⊗Hje
−V ,
where Kn =
∑n−1
k=0
1
νk
Hk ⊗ Hk for νk =
∫
RH
2
ke
−2V dx. By the classical Christoffel–Darboux
formula for orthogonal polynomials and the three term recurrence relation, it follows that
(x− y)Sn(x, y) has rank at most D + 1. Indeed, this is one of the ideas behind Widom’s
method.
4 A dynamical system
This paper is part of a broader project to develop structures for β = 1, 4 random matrix
ensembles parallel to those of β = 2 ensembles. A central part of the β = 2 theory is the
relation between orthogonal polynomials and the celebrated Toda lattice (see [14, Chapter 2]).
One way to derive this relation is, starting with the Fokas–Its–Kitaev Riemann–Hilbert problem
for orthogonal polynomials [27], to pass to a problem with constant jump matrices and then by
standard techniques of varying parameters one derives a Lax pair whose compatibility condition
gives the Toda lattice in Flaschka variables. In this section we repeat this method to see what
analogous dynamical system arises for our Riemann–Hilbert Problem 2.7. The calculation is
complicated and so we confine ourselves simply to deriving the compatibility condition, and
leave the question of implications for the theory of integrable systems to future research.
We begin, analogously to the Toda lattice, by introducing a t ∈ R dependence into the
potential as
V (z, t) := V0(z) + tz. (4.1)
Let Ân(·, t) be the solution of Riemann–Hilbert Problem 2.24 with potential V (·, t) and let
Ân(·, t) have the following expansion at z = ∞:
Ân(z, t) =
(
I+ z−1Â(1)
n (t) + z−2Â(2)
n (t) +O
(
z−3
))
z−2n−D+1
z2n
z
. . .
z
,
z → ∞.
Introduce the variables a
(1)
ij (n, t)
def
= Â
(1)
n (t)ij and a
(2)
ij (n, t)
def
= Â
(2)
n (t)ij for i, j ∈ [[1, D + 1]]. In
what follows, we shall often not write the dependence on t since this is always understood.
Furthermore, introduce the variable
bj(n, t)
def
=
D+1∑
k=3
a
(1)
2k (n, t)a
(1)
kj (n, t), j ∈ [[3, D + 1]]. (4.2)
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 25
Theorem 4.1. The matrix elements a
(1)
22 , a
(1)
2i , a
(1)
i2 , a
(2)
22 , a
(2)
12 , a
(2)
i2 and bi, for i = 3, . . . , D+ 1,
satisfy the following first-order dynamical system (n ≥ 0, t ∈ R):
da
(1)
22 (n)
dt
=
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n),
da
(1)
2i (n)
dt
= bi(n), i ∈ [[3, D + 1]],
da
(1)
i2 (n)
dt
= a
(2)
i2 (n)− a
(1)
i2 (n)a
(1)
22 (n), i ∈ [[3, D + 1]],
da
(2)
22 (n)
dt
=
D+1∑
k=3
a
(1)
2k (n)a
(2)
k2 (n),
d
dt
ln a
(2)
12 (n+ 1) = 2a
(1)
22 (n)−
D+1∑
j=3
a
(1)
2j (n)a
(2)
j2 (n+ 1),
d
dt
a
(2)
i2 (n+ 1) = a
(2)
i2 (n+ 1)a
(1)
22 (n) + a
(1)
i2 (n+ 1)a
(2)
22 (n) + a
(1)
i2 (n)− a
(1)
i2 (n+ 1)a
(2)
22 (n+ 1)
+ a
(1)
i2 (n+ 2)
a
(2)
12 (n+ 1)
a
(2)
12 (n+ 2)
− a
(1)
i2 (n+ 1)a
(1)
22 (n)
2
− a
(1)
i2 (n+ 1)
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n), i ∈ [[3, D + 1]],
and for n ≥ 1 and t ∈ R,
dbi(n)
dt
=
a
(2)
12 (n)
a
(2)
12 (n+ 1)
a
(1)
2i (n− 1) + a
(1)
22 (n+ 1)a
(1)
22 (n)a
(1)
2i (n) + a
(1)
2i (n+ 1)
− a
(2)
22 (n+ 1)a
(1)
2i (n) + a
(2)
22 (n)a
(1)
2i (n)− a
(1)
2i (n)
(
a
(1)
22 (n)
)2 − a
(1)
22 (n)bi(n)
− 2a
(1)
2i (n)
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n) + a
(1)
22 (n+ 1)bi(n), i ∈ [[3, D + 1]].
Note that a
(2)
12 (n+ 1) ̸= 0 so the above is well defined. The above dynamics are furthermore
constrained by the relation
D+1∑
j=3
a
(1)
2j (n)a
(1)
j2 (n+ 1) = 2. (4.3)
Our “initial data” with respect to n must satisfy a
(1)
22 (0, t) = a
(1)
i2 (0, t) = a
(2)
22 (0, t) = a
(2)
12 (0, t) =
a
(2)
i2 (0, t) = bi(0, t) = 0 for i ∈ [[3, D + 1]].
The identity (4.3) may be seen as analogous to (3.3). What follows in the rest of this section
is a proof of Theorem 4.1.
Remark 4.2. Note that repeated differentiation of (4.3) yields an infinite number of identities,
so that the k-th derivative of (4.3) yields an identity involving matrix elements at k different
values of n.
Remark 4.3. Some of the matrix elements not included in Theorem 4.1 may be related to those
included by the following identities:
a
(1)
i1 (n) = a
(2)
21 (n)a
(1)
i2 (n+ 1), i ∈ [[3, D + 1]], (4.4)
a
(1)
1i (n+ 1) = a
(2)
12 (n+ 1)a
(1)
2i (n), i ∈ [[3, D + 1]], (4.5)
1 = a
(2)
21 (n)a
(2)
12 (n+ 1), 0 = a
(1)
12 (n) = a
(1)
21 (n) = a
(2)
ij (n),
i ∈ [[1, D + 1]], j ∈ [[3, D + 1]].
26 A. Little
These identities can be seen directly from the solution, though (4.4) and (4.5) will be deduced
in the course of proving Theorem 4.1. The matrix elements a
(2)
i1 for i ̸= 2 are left undetermined
by these relations. Furthermore, a
(1)
ij (n, t), for i, j ≥ 3 do not enter into the dynamical system
on their own but only in the combination of the variable bi(n, t) (4.2).
Example 4.4. In the case D = 2, it is possible to give an exact solution. Without loss of
generality (by shifting and rescaling), one can take V0(z) =
1
2z
2. With this initial condition one
finds the following:
Â(1)
n (t) =
−t(2n+ 1) 0 2πie−
1
2 t2
√
2(n− 1
2)!
0 2nt − e
1
2 t2n!√
2
e
1
2 t22
√
2(n+ 1
2)!
2πi −2
√
2e−
1
2 t2
(n−1)! t
,
Â(2)
n (t) =
∗ −i
√
π 22ne−t2
(2n−1)! 0
− (2n+1)!
√
π
22n+1
et
2
2πi n(2n− 1)
(
t2 − 1
2
)
+ n 0
∗ −2
√
2(2n−1)te−
1
2 t2
(n−1)! 0
.
We find also b3(n, t) = − te
1
2 t2n!√
2
. The reader may verify that this is an exact solution of the
dynamical system in Theorem 4.1 for the case D = 2.
Let us introduce the (D + 1)× (D + 1) matrices
E1 = diag(1, 0, . . . , 0︸ ︷︷ ︸
D times
), E2 = diag(0, 1, 0, . . . , 0︸ ︷︷ ︸
D−1 times
), σ = E1 − E2.
Let us then define
(
A n(z, t)
def
= Ân(z, t)e
V (z,t)σ.
It is then easy to see that
(
A n satisfies the following Riemann–Hilbert problem.
Riemann–Hilbert Problem 4.5. Let Γ be as in (2.4). Then
(
A n has the following properties:
(1) z 7→
(
A n(z, t) is an analytic function on C \ Γ.
(2) z 7→
(
A n(z, t) has continuous left (+) and right (−) non-tangential boundary valuesx 7→
(
A±
n (x, t) for x ∈ Γ \ {0} which are related by the jump condition
(
A+
n (x, t) =
(
A−
n (x, t)
1
−χR 1
−χΓ1 1
...
. . .
−χΓD−1
1
, x ∈ Γ \ {0}.
(3) For each t ∈ R, the function z 7→
(
A n(z, t) is bounded in a neighbourhood of 0 ∈ C.
(4) For each t ∈ R, z 7→
(
A n(z, t) satisfies the asymptotic normalisation
(
A n(z, t) =
(
I+O
(
z−1
))
z−2n−D+1
z2n
z
. . .
z
eV (z,t)σ, z → ∞.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 27
Because the jump matrix is independent of both n and t, standard Riemann–Hilbert theory
now enables us to derive a dynamical system. In particular
(
A n+1(z, t) and
∂
(
A n(z,t)
∂t both satisfy
the first to third properties, differing only on the fourth property. Because of the continuity of
the boundary values and boundedness at 0, we find that
z 7→
(
A n+1(z, t)
(
A n(z, t)
−1, z 7→ ∂
(
A n(z, t)
∂t
(
A n(z, t)
−1
extend to entire functions on C. Let us write the asymptotic expansion (z → ∞)
(
A n(z, t) =
(
I+ z−1Â(1)
n (t) + z−2Â(2)
n (t) +O
(
z−3
))
z−2n−D+1
z2n
z
. . .
z
eV (z,t)σ.
A short calculation yields the Lax pair,
(
A n+1(z, t) =
(
z2E2 + zρ(1)n (t) + ρ(2)n (t)
) (
A n(z, t), (4.6)
∂
(
A n(z, t)
∂t
= (zσ + κn(t))
(
A n(z, t), (4.7)
where
ρ(1)n (t) = Â
(1)
n+1(t)E2 − E2Â
(1)
n (t),
ρ(2)n (t) = I− E1 − E2 + Â
(2)
n+1(t)E2 + E2
(
Â(1)
n (t)2 − Â(2)
n (t)
)
− Â
(1)
n+1(t)E2Â
(1)
n (t),
κn(t) = [Â(1)
n (t), σ].
Remark 4.6. The reader will observe that the jump matrices of
(
A n are not only independent
of n and t but also independent of z. Thus one could also derive (n, z) and (t, z) Lax pairs.
Indeed, one could consider more general deformations than (4.1).
Remark 4.7 (analogue of three term recurrence). We note that (4.6) implies a similar equation
for Ân, i.e.,
Ân+1(z) =
(
z2E2 + zρ(1)n + ρ(2)n
)
Ân(z).
Considering the second column of both sides yields a recursion relation, quadratic in z, which
may be regarded as a β = 4 analogue of the three term recurrence for orthogonal polynomials
(recall equations (2.7), (2.8), (2.9)).
Before proceeding, we must indicate a point of rigour. To compute this Lax pair we have
exchanged a z → ∞ limit with a derivative with respect to t. How can this be justified? The
proof is elementary but tedious to write out and so we only sketch how this is done. The solution
of our even problem Ân is expressible as rational combinations of integrals of the form∫
Γj
xke−V (x,t)dx,
∫
R
xke−2V (x,t)dx,
∫
Γj
xk
x− z
e−V (x,t)dx,
∫
R
xk
x− z
e−2V (x,t)dx.
The proof reduces to showing that such integrals are C∞ functions of t and that we can differ-
entiate under the integral sign. Both can be justified by a dominated convergence argument.
This implies that Â
(1)
n and Â
(2)
n are C∞ functions of t.
28 A. Little
Moving forward, let us write
ρ(z, t, n) = z2E2 + zρ(1)n (t) + ρ(2)n (t), λ(z, t, n) = zσ + κn(t).
Then the compatibility condition (or zero curvature condition) of the above Lax pair (4.6)
and (4.7) is
∂ρ(z, t, n)
∂t
+ ρ(z, t, n)λ(z, t, n)− λ(z, t, n+ 1)ρ(z, t, n) = 0, ∀z ∈ C. (4.8)
The left-hand side is at most a cubic polynomial in z, all of whose coefficients must vanish. In
fact the coefficient of z3 is [E2, σ] = 0 which yields no information. The coefficient of z2 yields
the equation
E2κn(t)− κn+1(t)E2 +
[
ρ(1)n (t), σ
]
= 0.
However, when we substitute in our formulas for ρ
(1)
n and κn we find that the left-hand side is
identically zero. So the quadratic coefficient also yields no information. Thus the left-hand side
of (4.8) is in fact a linear polynomial in z.
The coefficient of z in (4.8) yields
dρ
(1)
n (t)
dt
+ ρ(1)n (t)κn(t)− κn+1(t)ρ
(1)
n (t) +
[
ρ(2)n (t), σ
]
= 0, (4.9)
while the constant term gives
dρ
(2)
n (t)
dt
+ ρ(2)n (t)κn(t)− κn+1(t)ρ
(2)
n (t) = 0. (4.10)
Equation (4.9), after substituting our expressions for ρ
(1)
n (t) and ρ
(2)
n (t), yields
−Â(2)
n+1E2 + Â
(1)
n+1E2Â
(1)
n+1E2 − E1Â
(2)
n+1E2 − E2Â
(2)
n + E2Â
(2)
n E2 − E2Â
(2)
n E1 + E2Â
(2)
n+1E2
− E2
dÂ
(1)
n
dt
− E2
(
Â
(1)
n+1
)2
E2 − E2Â
(1)
n E2Â
(1)
n + E2
(
Â(1)
n
)2
+
dÂ
(1)
n+1
dt
E2 + E1
(
Â
(1)
n+1
)2
E2
= 0. (4.11)
The (1, 1) matrix element of (4.11) is 0. The (1, 2) matrix element yields the identity
2a
(2)
12 (n) =
D+1∑
k=3
a
(1)
1k (n)a
(1)
k2 (n). (4.12)
The (2, 1) matrix element yields
2a
(2)
21 (n) =
D+1∑
k=3
a
(1)
2k (n)a
(1)
k1 (n). (4.13)
The (2, 2) element gives that the quantity
(n, t) 7→ d
dt
a
(1)
22 (n, t)−
D+1∑
k=3
a
(1)
2k (n, t)a
(1)
k2 (n, t)
is independent of n.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 29
Lemma 4.8.
d
dt
a
(1)
22 (n) =
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n).
Proof. To show this, we need only show that
d
dt
a
(1)
22 (0, t)−
D+1∑
k=3
a
(1)
2k (0, t)a
(1)
k2 (0, t) = 0.
However, if n = 0, then
(
Ân
)
22
= P2n = P0 = 1, and hence a
(1)
22 (0, t) = 0. Similarly,
(
Ân
)
k2
= 0
for k ≥ 3, and so a
(1)
k2 (0, t) = 0. ■
The (1, i) and (i, 1) matrix elements for i ≥ 3 give 0. The (2, i) matrix element for i ≥ 3 gives
da
(1)
2i (n)
dt
=
D+1∑
k=3
a
(1)
2k (n)a
(1)
ki (n), i ∈ [[3, D + 1]].
Finally, the (i, 2) matrix element for i ≥ 3 gives the equation
da
(1)
i2 (n, t)
dt
= a
(2)
i2 (n, t)− a
(1)
i2 (n, t)a
(1)
22 (n, t), i ∈ [[3, D + 1]].
Next let us move onto the constant term (4.10), which gives us the complicated expression
E2Â
(2)
n E1Â
(1)
n − E2Â
(2)
n Â(1)
n E1 + E2Â
(2)
n Â(1)
n E2 − E2Â
(1)
n+1Â
(2)
n+1E2 + E2Â
(1)
n+1E2
− E2
(
Â(1)
n
)3
E2 − Â
(1)
n+1E2Â
(1)
n E2Â
(1)
n − Â
(2)
n+1E2Â
(1)
n E2 + E2Â
(1)
n+1E2Â
(2)
n
+ E1Â
(1)
n+1Â
(2)
n+1E2 − Â
(1)
n+1E2
(
Â(1)
n
)2
E1 + E2Â
(1)
n
dÂ
(1)
n
dt
− E1
(
Â
(1)
n+1
)2
E2Â
(1)
n
− E2
dÂ
(2)
n
dt
− E2Â
(1)
n+1E2
(
Â(1)
n
)2 − E1Â
(1)
n E1 + E2
dÂ
(1)
n
dt
Â(1)
n − Â
(1)
n+1E2
dÂ
(1)
n
dt
+
dÂ
(2)
n+1
dt
E2 − E2Â
(1)
n+1 − E2
(
Â(1)
n
)2
E1Â
(1)
n + E2
(
Â
(1)
n+1
)2
E2Â
(1)
n + Â
(2)
n+1E2Â
(1)
n
+ Â(1)
n E1 − Â
(1)
n+1E2Â
(2)
n − Â(1)
n E2 + E2Â
(1)
n E2 + E2
(
Â(1)
n
)2
E2Â
(1)
n
− Â
(1)
n+1E2Â
(1)
n+1E2Â
(1)
n −
dÂ
(1)
n+1
dt
E2Â
(1)
n + Â
(1)
n+1E2Â
(2)
n+1E2 + E2
(
Â(1)
n
)3
E1 + E1Â
(1)
n+1
+ Â
(1)
n+1E2
(
Â(1)
n
)2
E2 + Â
(1)
n+1E2
(
Â(1)
n
)2 − E2Â
(2)
n E2Â
(1)
n − E1Â
(1)
n+1E1
− Â
(1)
n+1E1Â
(2)
n+1E2 = 0. (4.14)
The (1, 1) matrix element of (4.14) is 0. The (2, 2) matrix element yields, after some simplifica-
tion, the relation
da
(2)
22 (n+ 1)
dt
−
D+1∑
k=3
a
(1)
2k (n+ 1)a
(2)
k2 (n+ 1) =
da
(2)
22 (n)
dt
−
D+1∑
k=3
a
(1)
2k (n)a
(2)
k2 (n).
Lemma 4.9.
da
(2)
22 (n)
dt
=
D+1∑
k=3
a
(1)
2k (n)a
(2)
k2 (n).
30 A. Little
Proof. Follows by the same observations as in Lemma 4.8. ■
The (2, 1) matrix element of (4.14) gives the following equation:
2
(
a
(1)
22 (n)− a
(1)
22 (n+ 1)
)
a
(2)
21 (n) +
d
dt
a
(2)
21 (n) +
D+1∑
j=3
bj(n)a
(1)
j1 (n) = 0. (4.15)
The (1, 2) matrix element gives
d
dt
a
(2)
12 (n+ 1) +
D+1∑
k=3
a
(1)
1k (n+ 1)a
(2)
k2 (n+ 1)− 2a
(1)
22 (n)a
(2)
12 (n+ 1) = 0. (4.16)
The (i, 1) matrix element for i ≥ 3 gives
a
(1)
i1 (n) = a
(2)
21 (n)a
(1)
i2 (n+ 1), i ∈ [[3, D + 1]]. (4.17)
The (1, i) matrix element gives
a
(1)
1i (n+ 1) = a
(2)
12 (n+ 1)a
(1)
2i (n), i ∈ [[3, D + 1]]. (4.18)
The (i, 2) matrix element yields after some simplification
d
dt
a
(2)
i2 (n+ 1) = a
(2)
i2 (n+ 1)a
(1)
22 (n) + a
(1)
i2 (n+ 1)a
(2)
22 (n) + a
(1)
i2 (n)− a
(1)
i2 (n+ 1)a
(2)
22 (n+ 1)
− a
(1)
i2 (n+ 1)
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n) + a
(2)
21 (n+ 1)a
(1)
i2 (n+ 2)a
(2)
12 (n+ 1)
− a
(1)
i2 (n+ 1)a
(1)
22 (n)
2, i ∈ [[3, D + 1]].
The (2, i) matrix element gives
−a(2)21 (n)a
(2)
12 (n)a
(1)
2i (n− 1)− a
(1)
22 (n+ 1)a
(1)
22 (n)a
(1)
2i (n)− a
(1)
2i (n+ 1) + a
(2)
22 (n+ 1)a
(1)
2i (n)
+ a
(1)
2i (n)
(
a
(1)
22 (n)
)2
+ 2a
(1)
2i (n)
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n) +
dbi(n)
dt
+
(
a
(1)
22 (n)− a
(1)
22 (n+ 1)
)
bi(n)− a
(2)
22 (n)a
(1)
2i (n) = 0, i ∈ [[3, D + 1]].
Let us now combine our formulas. Substituting (4.17) into (4.13), we find
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n+ 1) = 2.
We would have found the same if we had substituted (4.18) into (4.12). Note that if we com-
bine (4.15) and (4.16), we find
d
dt
ln a
(2)
21 (n) +
d
dt
ln a
(2)
12 (n+ 1)
= 2a
(1)
22 (n+ 1)−
D+1∑
k=3
bk(n)a
(1)
k2 (n+ 1)−
D+1∑
k=3
a
(1)
2k (n)a
(2)
k2 (n+ 1)
= − d
dt
D+1∑
k=3
a
(1)
2k (n)a
(1)
k2 (n+ 1)︸ ︷︷ ︸
=2
= 0,
which we already knew since a
(2)
21 (n)a
(2)
12 (n+ 1) = 1. Finally, the (i, j) matrix element of (4.14)
for i, j ≥ 3, after substituting the identities we have derived so far, is identically zero so yields
no further information. This completes the proof of Theorem 4.1.
A Riemann–Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type 31
Acknowledgements
The author would like to express his gratitude to Thomas Bothner and Mattia Cafasso, whose
comments and suggestions have been invaluable for this paper. The author would also like to
express his gratitude to the anonymous referees whose comments have improved the presentation
of the paper and have drawn attention to connections to recent work. This work was supported
by the UK Engineering and Physical Sciences Research Council through grant EP/T013893/2
and by the European Research Council (ERC) under the European Union Horizon 2020 research
and innovation program (grant agreement No. 884584).
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1 Introduction
1.1 Notation
2 Riemann–Hilbert representations
3 Christoffel–Darboux type formula
4 A dynamical system
References
|
| id | nasplib_isofts_kiev_ua-123456789-212344 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T14:52:43Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Little, Alex 2026-02-05T09:54:17Z 2024 A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type. Alex Little. SIGMA 20 (2024), 076, 32 pages 1815-0659 2020 Mathematics Subject Classification: 60B20; 33C45; 30E15; 30E25 arXiv:2306.14107 https://nasplib.isofts.kiev.ua/handle/123456789/212344 https://doi.org/10.3842/SIGMA.2024.076 We present a representation of skew-orthogonal polynomials of symplectic type ( = 4) in terms of a matrix Riemann-Hilbert problem, for weights of the form e⁻ⱽ⁽ᶻ⁾ where is a polynomial of even degree and positive leading coefficient. This is done by representing skew-orthogonality as a kind of multiple-orthogonality. From this, we derive a = 4 analogue of the Christoffel-Darboux formula. Finally, our Riemann-Hilbert representation allows us to derive a Lax pair whose compatibility condition may be viewed as a = 4 analogue of the Toda lattice. The author would like to express his gratitude to Thomas Bothner and Mattia Cafasso, whose comments and suggestions have been invaluable for this paper. The author would also like to express his gratitude to the anonymous referees whose comments have improved the presentation of the paper and have drawn attention to connections to recent work. This work was supported by the UK Engineering and Physical Sciences Research Council through grant EP/T013893/2 and by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 884584). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type Article published earlier |
| spellingShingle | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type Little, Alex |
| title | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| title_full | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| title_fullStr | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| title_full_unstemmed | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| title_short | A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type |
| title_sort | riemann-hilbert approach to skew-orthogonal polynomials of symplectic type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212344 |
| work_keys_str_mv | AT littlealex ariemannhilbertapproachtoskeworthogonalpolynomialsofsymplectictype AT littlealex riemannhilbertapproachtoskeworthogonalpolynomialsofsymplectictype |