Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems
Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding syst...
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| Цитувати: | Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems / H. Miki, H. Goda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1483732019-02-19T01:28:47Z Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems Miki, H. Goda, H. Tsujimoto, S. Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2×2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations. 2012 Article Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems / H. Miki, H. Goda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 35C05; 37K60; 15B52 DOI: http://dx.doi.org/10.3842/SIGMA.2012.008 http://dspace.nbuv.gov.ua/handle/123456789/148373 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2×2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations. |
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Article |
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Miki, H. Goda, H. Tsujimoto, S. |
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Miki, H. Goda, H. Tsujimoto, S. Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Miki, H. Goda, H. Tsujimoto, S. |
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Miki, H. |
| title |
Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems |
| title_short |
Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems |
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Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems |
| title_fullStr |
Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems |
| title_full_unstemmed |
Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems |
| title_sort |
discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems |
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Інститут математики НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/148373 |
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Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems / H. Miki, H. Goda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT mikih discretespectraltransformationsofskeworthogonalpolynomialsandassociateddiscreteintegrablesystems AT godah discretespectraltransformationsofskeworthogonalpolynomialsandassociateddiscreteintegrablesystems AT tsujimotos discretespectraltransformationsofskeworthogonalpolynomialsandassociateddiscreteintegrablesystems |
| first_indexed |
2025-07-12T19:14:58Z |
| last_indexed |
2025-07-12T19:14:58Z |
| _version_ |
1837469748858716160 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 008, 14 pages
Discrete Spectral Transformations
of Skew Orthogonal Polynomials
and Associated Discrete Integrable Systems
Hiroshi MIKI, Hiroaki GODA and Satoshi TSUJIMOTO
Department of Applied Mathematics and Physics, Graduate School of Informatics,
Kyoto University, Sakyo-Ku, Kyoto 606 8501, Japan
E-mail: miki@amp.i.kyoto-u.ac.jp, tujimoto@i.kyoto-u.ac.jp
URL: http://www-is.amp.i.kyoto-u.ac.jp/lab/en/miki/
Received December 01, 2011, in final form February 20, 2012; Published online February 29, 2012
http://dx.doi.org/10.3842/SIGMA.2012.008
Abstract. Discrete spectral transformations of skew orthogonal polynomials are presented.
From these spectral transformations, it is shown that the corresponding discrete integrable
systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-
dimensional case, the corresponding system can be extended to 2 × 2 matrix form. The
factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random
matrix theory is presented as a by-product of these transformations.
Key words: skew orthogonal polynomials; discrete integrable systems; discrete coupled KP
equation; Pfaff lattice; Christoffel–Darboux kernel
2010 Mathematics Subject Classification: 42C05; 35C05; 37K60; 15B52
1 Introduction
Orthogonal polynomials and their varieties have been investigated for a long time and yielded
a lot of contributions to various areas including quantum physics, random matrices, wavelets
and so on. Over the last twenty years, several integrable systems, both in continuous variable
and in discrete variable, have been found to relate to such polynomials through their spectral
transformations. One of the well-known results is the relation between ordinary orthogonal
polynomials (OPs) and Toda chain (see [5, 21] for further details). For recent years, from this
point of view, new types of integrable systems have been derived [23, 22] and the algebraic
structures of their solutions also have been discussed in [19, 18].
In this paper, we treat the skew orthogonal polynomials (SOPs), first introduced in the theo-
ry of random matrices [7]. It is known that the continuous spectral transformation of SOPs
induces the semi-discrete integrable system, known as Pfaff lattice [2, 3], and its relations to the
Lie algebra, geometry and matrix eigenvalue problems are discussed in [2, 14, 15] respectively.
Although, the full-discrete integrable system associated with SOPs, namely the discrete coun-
terpart of Pfaff lattice, still remains to be found and even the discrete spectral transformation of
SOPs is not obtained. Recalling the application of the discrete integrable system to several areas
such as eigenvalue problems and so on (e.g. [17, 26]), the discrete integrable system associated
with SOPs, if obtained, can be expected to have an application or contribution to such areas.
We shall derive the corresponding full-discrete integrable systems from their discrete spectral
transformations.
This paper is organized as follows. In Section 2, a brief review of the relationship between OPs
and the discrete time Toda equation is given and we propose the discrete spectral transformation
of SOPs. From these transformations, we derive the corresponding discrete integrable system,
mailto:miki@amp.i.kyoto-u.ac.jp
mailto:tujimoto@i.kyoto-u.ac.jp
http://www-is.amp.i.kyoto-u.ac.jp/lab/en/miki/
http://dx.doi.org/10.3842/SIGMA.2012.008
2 H. Miki, H. Goda and S. Tsujimoto
which takes (1+1)-dimensional form. In Section 3, another discrete integrable system associated
with SOPs is shown to be derived also from their discrete spectral transformations, which takes
(2 + 1)-dimensional form. In Section 4, we also show that the system obtained in Section 3 can
be extended to 2× 2 matrix form. In Section 5, concluding remarks are presented.
2 Skew orthogonal polynomials
We first briefly review a relation between the discrete Toda time equation and the discrete
spectral transformation of OPs according to [21]. The discrete time Toda equation (on semi-
infinite lattice)
U tnV
t
n = U t+1
n−1V
t+1
n , U tn + V t
n+1 = U t+1
n + V t+1
n , V t
0 = 0,
is known to admit the following discrete Lax pair:
xΦt+1 =
U t0 1
U t1 1
U t2 1
. . .
. . .
Φt, Φt =
1
V t
1 1
V t
2 1
. . .
. . .
Φt+1, (1)
where x is a spectral parameter and Φt = (φt0(x), φt1(x), . . . )T is a wave function. From (1), we
obtain
xΦt+1 =
U t0 + V t
1 1
U t1V
t
1 U t1 + V t
2 1
U t2V
t
2 U t2 + V t
3 1
. . .
. . .
. . .
Φt+1,
for all t. If we normalize φt0(x) = 1, Favard’s theorem shows that {φtn(x)}∞n=0 is a sequence of
monic orthogonal polynomials, i.e. there exists some linear functional Lt : R[x]→ R such that
Lt[φtm(x)φtn(x)] = htnδmn, htn 6= 0.
Therefore, one can easily find that the discrete Lax pair (1) is just the discrete spectral trans-
formation of orthogonal polynomials known as Christoffel transformation and Geronimus trans-
formation [6, 8], respectively.
Our main aim is to derive the discrete integrable systems associated with SOPs. To that
end, we shall give the discrete spectral transformations of SOPs like Christoffel and Geronimus
transformation. We first begin with the definition of SOPs:
Definition 1. Let 〈·|·〉 be the bilinear 2-form from R[z] × R[z] to → R which satisfy the skew
symmetric relation
〈f(z)|g(z)〉 = −〈g(z)|f(z)〉.
This skew-symmetric bilinear 2-form is often called “skew-inner product”. The SOPs {qn(z)}∞n=0
with respect to the skew-inner product 〈·|·〉 are defined as the set of polynomials satisfying the
following relations:
〈q2m(z)|q2n+1(z)〉 = rnδmn, rn 6= 0,
〈q2m(z)|q2n(z)〉 = −〈q2m+1(z)|q2n+1(z)〉 = 0, (2)
for all nonnegative integers m, n.
Discrete Spectral Transformations of SOPs and Associated DIS 3
From now on, unless specified, we assume that SOPs are monic, which ensures the uniqueness
of SOPs of even degree. However, SOPs of odd degree have the following ambiguity: the skew
orthogonality relation (2) is invariant under the replacement
q2n+1(z)→ q2n+1(z) + αnq2n(z), (3)
for any αn ∈ R. In many cases, the coefficient of z2n in q2n+1(z) is usually chosen as 0 and then
the SOPs are uniquely determined.
The discrete spectral transformation of SOPs (that is, the transformation from SOPs to
another SOPs) can be constructed.
Theorem 1. Let {qn(z)}∞n=0 be SOPs with respect to the skew-inner product 〈·|·〉. Let us denote
some constant parameter by λ satisfying
λ ∈ {z ∈ C | q2n(z) 6= 0, for all n = 0, 1, . . . }.
Consider the new sequence of polynomials {q∗n(z)}∞n=0 defined by
q∗2n(z) =
1
z − λ
(
n∑
k=0
rn
rk
· q2k+1(z)q2k(λ)− q2k+1(λ)q2k(z)
q2n(λ)
)
, (4)
q∗2n+1(z) =
1
z − λ
(
q2n+2(z)−
q2n+2(λ)
q2n(λ)
q2n(z)
)
+ αnq
∗
2n(z), (5)
where αn is an arbitrary parameter. Then {q∗n(z)}∞n=0 are again SOPs with respect to the modified
skew-inner product
〈·|·〉∗ := 〈(z − λ) · |(z − λ)·〉. (6)
Proof. In this proof, because of the equation (3), we take the arbitrary parameter αn = 0
for all n in (5) for simplicity. In order to prove the theorem, it is enough to check the skew
orthogonal relation (2) for {q∗n(z)}∞n=0. Using the skew orthogonality relation (2) for {qn(z)}∞n=0,
we find
〈q∗2n(z), q∗2m+1(z)〉∗ = 〈(z − λ)q∗2n(z), (z − λ)q∗2m+1(z)〉
=
〈
n∑
k=0
q2k(λ)q2k+1(z)− q2k+1(λ)q2k(z)
rkr
−1
n q2n(λ)
∣∣∣∣∣ q2m+2(z)−
q2m+2(λ)
q2m(λ)
q2m(z)
〉
=
rn
q2n(λ)
n∑
k=0
〈
q2k(λ)q2k+1(z)
rk
∣∣∣∣ q2m+2(z)−
q2m+2(λ)
q2m(λ)
q2m(z)
〉
=
q2n+2(λ)
q2n(λ)
rnδmn.
Similarly, direct calculations also show us
〈q∗2m(z), q∗2n(z)〉∗ = 〈q∗2n+1(z), q
∗
2m+1(z)〉∗ = 0.
This completes the proof. �
Now observing the discrete spectral transformation of even-degree SOPs (4), one can easily
find that except for the multiplier factor, this is equivalent to the Christoffel–Darboux kernel
for skew orthogonal polynomials (we will call this “skew-Christoffel kernel”) [9, 16] introduced
in the theory of random matrices which takes the form
IN (x, y) =
N∑
k=0
q2k+1(x)q2k(y)− q2k+1(y)q2k(x)
rk
, (7)
where {qn(z)}∞n=0 are some SOPs. We hence obtain the following corollary:
4 H. Miki, H. Goda and S. Tsujimoto
Corollary 1. The skew-Christoffel kernel (7) can be factorized into the two SOPs:
IN (x, y) = (x− y)
q2N (x)q∗2N (x)
rN
, (8)
where the SOPs q2N (x) and q∗2N (x) are connected to each other by the spectral transformation (2)
with z = x, λ = y.
There exist many works on the skew-Christoffel kernel for the specific weight cases (see
e.g. [1, 9]). Here we stress that the factorization (8) holds not only for the case mentioned above
but also for any well-defined skew inner product. Especially, consider the skew inner product
in orthogonal and symplectic ensemble given by:
〈f(z)|g(z)〉orthogonal =
∫∫
D2
sgn(z1 − z2)f(z1)g(z2)w(z1)w(z2)dz1dz2,
〈f(z)|g(z)〉symplectic =
∫
D
(f(z)g′(z)− f ′(z)g(z))w(z)dz, (9)
where f ′(z) = d
dzf(z) and D, w(z) are the corresponding integral domain and weight function
respectively. For such skew inner product, the modified skew inner product (6) can be written
as follows:
〈(z − λ)f(z)|(z − λ)g(z)〉orthogonal =
∫∫
D2
sgn(z1 − z2)f(z1)g(z2)w̄(z1)w̄(z2)dz1dz2,
〈(z − λ)f(z)|(z − λ)g(z)〉symplectic =
∫
D
(f(z)g′(z)− f ′(z)g(z))w̄(z)dz,
with w̄(z) = (z − λ)w(z). This shows that the skew-Christoffel kernel in random matrix theory
can be factorized into two even-degree SOPs with respect to the different weight function.
Remark 1. Similar relations hold between the Christoffel transformation of OPs and the
Christoffel kernels. Hence we call the discrete spectral transformation (4), (5) as the skew-
Christoffel transformation.
Iterating the skew-Christoffel transformation, we obtain a sequence of SOPs by the following
procedure:
q0n(z) := qn(z),
qt+1
2n (z) :=
1
z − λ
(
qt2n+1(z) +
n∑
k=0
Atn,kq
t
2k(z) +
n−1∑
k=0
Bt
n,kq
t
2k+1(z)
)
,
qt+1
2n+1(z) :=
1
z − λ
(
qt2n+2(z) + Ctnq
t
2n(z)
)
+ αtnq
t+1
2n (z),
where
Atn,k =
rtn
rtk
·
qt2k(λ)
qt2n(λ)
, Bt
n,k = −r
t
n
rtk
·
qt2k+1(λ)
qt2n(λ)
, Ctn = −
qt2n+2(λ)
qt2n(λ)
.
Here we denote the corresponding skew-inner product by 〈·|·〉t. Then, skew-orthogonality rela-
tions are given as follows:
〈·|·〉0 := 〈·|·〉, 〈·|·〉t+1 := 〈(z − λ) · |(z − λ)·〉t,
〈qt2m(z), qt2n+1(z)〉t = rtnδmn, 〈qt2m(z), qt2n(z)〉t = 〈qt2m+1(z), q
t
2n+1(z)〉t = 0. (10)
For simplicity, we assume αtn = 0.
Discrete Spectral Transformations of SOPs and Associated DIS 5
We then proceed to find the inverse transformation of the skew-Christoffel transformation.
Using the skew-orthogonality relation (2), we can indeed construct the different contiguous
relations between {qtn(z)}∞n=0 and {qt+1
n (z)}∞n=0 as follows:
qt2n(z) = qt+1
2n (z) +
n−1∑
k=0
αtn,kq
t+1
2k (z) +
n−1∑
k=0
βtn,kq
t+1
2k+1(z),
qt2n+1(z) = qt+1
2n+1(z) +
n∑
k=0
γtn,kq
t+1
2k (z) +
n−1∑
k=0
εtn,kq
t+1
2k+1(z), (11)
where
αtn,k =
1
rt+1
k
〈(z − λ)qt+1
2n (z)|(z − λ)qt+1
2k+1(z)〉
t,
βtn,k =
1
rt+1
k
〈(z − λ)qt+1
2k (z)|(z − λ)qt+1
2n (z)〉t,
γtn,k =
1
rt+1
k
〈(z − λ)qt2n+1(z)|(z − λ)qt+1
2k+1(z)〉
t,
εtn,k =
1
rt+1
k
〈(z − λ)qt+1
2k (z)|(z − λ)qt2n+1(z)〉t.
Combining (10) and (11), we can form the following Lax pair:
(z − λ)Φt+1 = LtΦt :=
At0,0 1
Ct0 0 1
At1,0 Bt
1,0 At1,1 1
0 0 Ct1 0 1
...
. . .
. . .
. . .
. . .
. . .
Φt,
Φt = RtΦt+1 :=
1
γt0,0 1
αt1,0 βt1,0 1
γt1,0 εt1,0 γt1,1 1
...
. . .
. . .
. . .
. . .
Φt+1, (12)
where the wave-function Φt = (qt0(z), q
t
1(z), q
t
2(z), . . . )
T is the sequence of SOPs. From the
compatibility condition of (12), the discrete Lax equation is derived:
LtRt = Rt+1Lt+1. (13)
We can observe that if we take the suitable limit, the second equation of (10) goes to the relation:
d
dt
〈·|·〉 = 〈z · |·〉+ 〈·|z·〉,
which appears in the Pfaff lattice proposed by Adler et al. [2]. The discrete Lax equation (13)
thus can be considered to be the discrete counterpart of the Pfaff lattice.
3 Discrete integrable systems of SOPs in 2 + 1 dimension
In the previous section, we derive the discrete integrable system (13) from the discrete spectral
transformation of SOPs. However the system is non-local and therefore complicated. The
6 H. Miki, H. Goda and S. Tsujimoto
similar problem occurs in the case of ordinary biorthogonal polynomials and discrete two-
dimensional Toda (discrete KP) equation. This problem can be solved by introducing “another
time-evolution” [25]. Furthermore, observing the relation between skew-Christoffel kernel and
SOPs (8), one can realize that only even-degree SOPs appear. We hence expect that even-
degree SOPs play an crucial role in the corresponding discrete integrable systems. Due to
the guiding principles mentioned above, we shall construct the discrete integrable systems in
(2 + 1)-dimensional form associated with only even-degree SOPs.
Let 〈·|·〉s,t be a skew-inner product such that
〈·|·〉s+1,t = 〈(z − µ) · |(z − µ)·〉s,t, 〈·|·〉s,t+1 = 〈(z − λ) · |(z − λ)·〉s,t, (14)
where we assume λ 6= µ. We denote the corresponding SOPs by {qs,tn (z)}∞n=0. From Theorem 1,
{qs,tn (z)}∞n=0 and {qs+1,t
n (z)}∞n=0 are connected by the relation (4) and (5) with λ = µ. Similarly,
{qs,tn (z)}∞n=0 are also related to {qs,t+1
n (z)}∞n=0 by (4) and (5).
We shall find the local contiguous relations among {qs,tn (z)}∞n=0. To that end, we first recall
the Pfaffian expression of ordinary SOPs [2], which will play a crucial role in the following
sections. Pfaffians are defined by:
Pf(i0, . . . , i2n−1) :=
∑
σ∈S2n
sgnσ
n!2n
∏
0≤i≤n−1
Pf(iσ(2i), iσ(2i+1)),
where the elements of Pfaffians Pf(i, j) are supposed to satisfy the skew-symmetric relation:
Pf(i, j) = −Pf(j, i). (15)
From the definition of Pfaffians, we can find the following properties of Pfaffians which resemble
those of determinants:
Pf(i0, . . . , i2n−1) =
2n−2∑
k=0
(−1)k(ik, i2n−1)(i0, . . . , îk, . . . , i2n−2), (16)
Pf(i0, . . . , a · ia + b · ib, . . . , i2n−1)
= aPf(i0, . . . , ia, . . . , i2n−1) + bPf(i0, . . . , ib, . . . , i2n−1), (17)
Pf(. . . , s, . . . , s, . . . ) = 0, (18)
where ĵ means the deletion of j and
Pf(i, a · ia + b · ib) := aPf(i, ia) + bPf(i, ib). (19)
For further details of Pfaffians, see e.g. [13, 20].
Employing the “skew” moments 〈zi|zj〉 as the elements of Pfaffians, SOPs can be written in
terms of Pfaffians as well as ordinary OPs in terms of determinants.
Proposition 1 ([2]). The SOPs with respect to 〈·|·〉 can be expressed in terms of Pfaffians:
q2n(z) =
Pf(0, 1, . . . , 2n, z)
Pf(0, 1, . . . , 2n− 1)
,
q2n+1(z) = q2n(z) =
Pf(0, 1, . . . , 2n− 1, 2n+ 1, z)
Pf(0, 1, . . . , 2n− 1)
+ αnq2n(z),
where αn is an arbitrary parameter and the elements of Pfaffians are defined by
Pf(i, j) = 〈zi|zj〉, Pf(i, z) = zi
for all nonnegative integers i, j.
Discrete Spectral Transformations of SOPs and Associated DIS 7
Extending this proposition, we have obtained the Pfaffian expressions of the time-evolved
even degree SOPs.
Theorem 2. The even degree SOPs {qs,t2n(z)}∞n=0,{qs+1,t
2n (z)}∞n=0,{qs,t+1
2n (z)}∞n=0, {qs+1,t+1
2n (z)}∞n=0
can be written in terms of the moments with respect to 〈·|·〉s,t:
qs,t2n(z) =
Pf(0, 1, . . . , 2n, z)
Pf(0, 1, . . . , 2n− 1)
,
qs+1,t
2n (z) =
1
z − µ
· Pf(0, 1, . . . , 2n+ 1, µ, z)
Pf(0, 1, . . . , 2n, µ)
,
qs,t+1
2n (z) =
1
z − λ
· Pf(0, 1, . . . , 2n+ 1, λ, z)
Pf(0, 1, . . . , 2n, λ)
,
qs+1,t+1
2n (z) =
1
(z − µ)(z − λ)
· Pf(0, 1, . . . , 2n+ 2, µ, λ, z)
Pf(0, 1, . . . , 2n+ 1, µ, λ)
, (20)
where the elements of Pfaffians are defined by
Pf(i, j) = ss,ti,j := 〈zi|zj〉s,t,
Pf(i, z) = zi, Pf(i, λ) = λi, Pf(i, µ) = µi,
Pf(z, µ) = Pf(µ, λ) = Pf(λ, z) = 0. (21)
Before proving this theorem, we show the properties of the Pfaffian elements.
Lemma 1. Employing the notation (19) in (17), for the elements of the Pfaffians defined
by (21), the following properties hold:
Pf((i+ 1)− µ · i, (j + 1)− µ · j) = Pf(i+ 1, j + 1)− µPf(i+ 1, j)− µPf(i, j) + µ2Pf(i, j),
Pf((i+ 1)− µ · i, µ) = 0.
Proof. Following the definition (19) and anti-symmetric property of Pfaffian elements (15),
one can find
Pf((i+ 1)− µ · i, (j + 1)− µ · j) = Pf((i+ 1)− µ · i, j + 1)− µPf((i+ 1)− µ · i, j)
= −Pf(j + 1, (i+ 1)− µ · i, ) + µPf(j, µ(i+ 1)− µ · i)
= −Pf(j + 1, i+ 1)− µPf(j + 1, i) + µPf(j, i+ 1)− µ2Pf(j, i)
= Pf(i+ 1, j + 1)− µPf(i+ 1, j)− µPf(i, j) + µ2Pf(i, j).
One can also find
Pf(i+ 1− µ · i, µ) = Pf(i+ 1, µ)− µPf(i, µ) = µi+1 − µi+1 = 0.
This completes the proof. �
Proof of Theorem 2. We introduce the function τ s,tn and τ̂ s,tn as follows:
τ s,tn = Pf(0, . . . , 2n− 1), τ̂ s,tn = Pf(0, . . . , 2n, z). (22)
It is also convenient to introduce the shift operator Ts and Tt satisfying
Ts[f(s)] = f(s+ 1), Tt[f(t)] = f(t+ 1)
and the notation
ab := a− b.
8 H. Miki, H. Goda and S. Tsujimoto
From Proposition 1 and the relation (14), It is easy to see
qs,t2n(z) =
τ̂ s,tn
τ s,tn
(23)
and
Ts[Pf(i, j)] = Pf(i+ 1, j + 1)− µPf(i+ 1, j)− µPf(i, j + 1) + µ2Pf(i, j),
Tt[Pf(i, j)] = Pf(i+ 1, j + 1)− λPf(i+ 1, j)− λPf(i, j + 1) + λ2Pf(i, j). (24)
With the help of (17) and (18), we can see
Pf(0, . . . , 2n, µ) = Pf(0, . . . , 2n, µ)− µPf(0, . . . , 2n− 1, 2n− 1, µ)
= Pf(0, . . . , 2n− 1, 2̃n− 1, µ),
where j̃ denotes (j + 1)− µ · j in the sense of (17). Iterating this transformation, we can find
Pf(0, . . . , 2n, µ) = Pf(0, 0̃, 1̃, . . . , 2̃n− 1, µ). (25)
Meanwhile, we can also find
Pf(0, ĩ0, . . . , ĩ2n−1, µ)
= Pf(ĩ0, . . . , ĩ2n−1)−
2n−1∑
k=0
(−1)kPf(ĩk, µ)Pf(0, ĩ0, . . . ,
̂̃
ik, . . . , ĩ2n−1, µ), (26)
where we have used the Pfaffian expansion (16). With the help of Lemma 1 and (24), it is
straightforward to see
Pf(̃i, j̃) = Ts[Pf(i, j)], Pf(̃i, µ) = 0.
Therefore, combining (25) and (26), we arrive at
Ts[Pf(0, . . . , 2n− 1)] = Pf(0, . . . , 2n, µ).
In a similar manner, by using Pf(̃i, λ) = (λ− µ)λi, we further obtain
Pf(0, . . . , 2n+ 1, µ, λ) = λµTs[Pf(0, . . . , 2n, λ)].
Finally, arranging these results, we have
τ s+1,t
n = Pf(0, . . . , 2n, µ), τ s,t+1
n = Pf(0, . . . , 2n, λ),
τ s+1,t+1
n = λ−1µ Pf(0, . . . , 2n+ 1, µ, λ). (27)
From these results, we can easily see that τ̂ s,tn is equal to τ s+1,t
n with the substitution z = λ.
Hence, in a similar manner, we can also obtain
τ̂ s+1,t
n = z−1µ Pf(0, . . . , 2n+ 1, µ, z), τ̂ s,t+1
n = z−1λ Pf(0, . . . , 2n+ 1, λ, z),
τ̂ s+1,t+1
n = z−1µ z−1λ λ−1µ Pf(0, . . . , 2n+ 2, µ, λ, z). (28)
Finally, from (23), (27) and (28), we obtain the relation (20). This completes the proof. �
Using Theorem 2, we can obtain the following contiguous relations of even-degree SOPs.
Discrete Spectral Transformations of SOPs and Associated DIS 9
Theorem 3. The series of even degree SOPs {qs,t2n(z)}∞n=0 satisfy the following contiguous rela-
tions:
zλq
s,t+1
2n (z)− zµqs+1,t
2n (z) = zµ,λA
s,t
n q
s+1,t+1
2n−2 (z)−Bs,t
n qs,t2n(z),
zµ,λq
s+1,t+1
2n (z)− q2n+2(z) = zλC
s,t
n qs,t+1
2n (z)− zµDs,t
n q
s+1,t
2n , (29)
with
As,tn = (µ− λ)
τ s,tn+1τ
s+1,t+1
n−1
τ s+1,t
n τ s,t+1
n
, Bs,t
n = (µ− λ)
τ s,tn τ s+1,t+1
n
τ s+1,t
n τ s,t+1
n
,
Cs,tn = (µ− λ)−1
τ s+1,t
n+1 τ s,t+1
n
τ s,tn+1τ
s+1,t+1
n
, Ds,t
n = (µ− λ)−1
τ s,t+1
n+1 τ s+1,t
n
τ s,tn+1τ
s+1,t+1
n
, (30)
where τ s,tn is defined in (22).
Proof. Using the identities of Pfaffians [13]:
Pf(i0, . . . , i2n−1, a, b, c, d)Pf(i0, . . . , i2n−1) = Pf(i0, . . . , i2n−1, a, b)Pf(i0, . . . , i2n−1, c, d)
− Pf(i0, . . . , i2n−1, a, c)Pf(i0, . . . , i2n−1, b, d)
+ Pf(i0, . . . , i2n−1, a, d)Pf(i0, . . . , i2n−1, b, c) (31)
and
Pf(i0, . . . , i2n, a, b, c)Pf(i0, . . . , i2n, d) = Pf(i0, . . . , i2n, a, b, d)Pf(i0, . . . , i2n, c)
− Pf(i0, . . . , i2n, a, c, d)Pf(i0, . . . , i2n, b)
+ Pf(i0, . . . , i2n, b, c, d)Pf(i0, . . . , i2n, a), (32)
we obtain the relation between τ s,tn and τ̂ s,tn :
zµ,λλµτ
s,t
n+1τ̂
s+1,t+1
n = zλτ
s+1,t
n+1 τ̂ s,t+1
n − zµτ s,t+1
n+1 τ̂ s+1,t
n + λµτ̂
s,t
n+1τ
s+1,t+1
n ,
λµτ
s+1,t+1
n τ̂ s,tn = zµτ
s,t+1
n τ̂ s+1,t
n − zλτ s+1,t
n τ̂ s,t+1
n + zµ,λτ
s,t
n+1τ̂
s+1,t+1
n−1 , (33)
where zµ,λ = zµzλ. Hence, from Theorem 2, we can easily arrive at the result. �
Observing Theorem 3, we can regard the relation (29) as the discrete Lax pair whose wave
function is the sequence of even-degree SOPs. Then we can directly obtain the following system.
Theorem 4. From the compatibility condition of (29), we can derive the following nonlinear
difference system:
As+1,t+1
n −As,tn+1 +Bs,t
n+1 −B
s+1,t+1
n = Cs,t+1
n − Cs+1,t
n +Ds+1,t
n −Ds,t+1
n ,
As+1,t
n Cs+1,t
n−1 = As,tn C
s,t
n , As,t+1
n Ds,t+1
n−1 = As,tn D
s,t
n ,
Bs+1,t
n Ds+1,t
n = Bs,t
n+1D
s,t
n , Bs,t+1
n Cs,t+1
n = Bs,t
n+1C
s,t
n . (34)
This system has a special solution which can be written in terms of Pfaffians as (30).
We can consider the system (34) as a discrete integrable system since this admits a discrete
Lax pair by construction and we can put plenty arbitrary parameters into the solution (30).
From the point of view of the τ -function τ s,tn , the relation (24) which the elements of the
Pfaffian satisfy is the same as the dispersion relation of the discrete coupled KP (dcKP) equa-
tion [10] and indeed one can notice that the equation (33) is just the same as the dcKP equation.
Under the choice of z as the spectral parameter, the dcKP equation yields the discrete Lax
pair (29). Therefore the system (34) can be regarded as the nonlinear form of the dcKP.
10 H. Miki, H. Goda and S. Tsujimoto
4 An extension to 2 × 2 matrix form
In the previous section, we propose the discrete integrable system in (2+1)-dimensional form (34)
from the discrete spectral transformation of the SOPs. This system, by construction, is related
only to the even-degree SOPs and not to odd-degree SOPs. Taking it into account that the
integrable systems associated with both even and odd degree SOPs were already derived [1, 27],
we can expect that there should exist a discrete integrable system in 2+1 dimension which is also
connected to the discrete spectral transformations of both even and odd degree SOPs. In order
to construct such system, as is described in the previous section, the ambiguity of odd-degree
SOPs (3) is the barrier. Conversely, in this section, we shall construct such system not avoiding
the ambiguity of odd-degree SOPs but utilizing this.
Using the skew-inner product 〈·|·〉s,t with the property (14), we introduce the new sequence
of polynomials {φs,tn (z)}∞n=0:
φs,t2n(z) =
τ̂n
s,t
σs,tn
, φs,t2n+1(z) =
σ̂s,tn
τ s,tn
, (35)
where σs,tn and σ̂s,tn are defined by
σs,tn := Pf(0, . . . , 2n− 2, 2n) + (sµ+ tλ)τ s,tn ,
σ̂s,tn := Pf(0, . . . , 2n− 1, 2n+ 1, z) + (sµ+ tλ)τ̂ s,tn , (36)
respectively. Recall that τ s,tn and τ̂ s,tn are defined by (22). here we remark that φs,t2n(z) are
no longer monic except for s = t = 0. Nevertheless, from Proposition 1, one can easily find
{φs,tn (z)}n=0 are also SOPs with respect to 〈·|·〉s,t.
As is seen in the previous section, in order to derive the discrete Lax pair of even-degree SOPs,
that is the relation between τ s,tn and τ̂ s,tn , the equation (33) plays a central role. According to this
observation, we wish to acquire the relation between τ s,tn , τ̂ s,tn , σs,tn and σ̂s,tn . First we examine
the properties of σs,tn and σ̂s,tn .
Lemma 2. The function σs,tn and σ̂s,tn defined in (36) hold the following properties:
σs+1,t
n = Pf(0, . . . , 2n− 1, 2n+ 1, µ) + (sµ+ tλ)τ s+1,t
n ,
σs,t+1
n = Pf(0, . . . , 2n− 1, 2n+ 1, λ) + (sµ+ tλ)τ s,t+1
n ,
σs+1,t+1
n = λ−1µ Pf(0, . . . , 2n, 2n+ 2, µ, λ) + (sµ+ tλ)τ s+1,t+1
n ,
σ̂s+1,t
n = z−1µ Pf(0, . . . , 2n, 2n+ 2, µ, z) + (sµ+ tλ)τ̂ s+1,t
n ,
σ̂s,t+1
n = z−1λ Pf(0, . . . , 2n, 2n+ 2, λ, z) + (sµ+ tλ)τ̂ s,t+1
n ,
σ̂s+1,t+1
n = λ−1µ z−1µ,λPf(0, . . . , 2n+ 1, 2n+ 3, µ, λ, z) + (sµ+ tλ)τ̂ s+1,t+1
n .
Proof. In a similar manner to the proof of Theorem 2, we can see
Pf(0, . . . , k̂ + 1, . . . , 2n+ 1, µ) = Pf(0, . . . , k, k + 2, k̃ + 2, . . . , 2̃n, µ)
= Pf(0, . . . , k, k + 2, k̃ + 2, . . . , 2̃n, µ)− µ2Pf(0, . . . , k, k, k̃ + 2, . . . , 2̃n, µ)
= Pf(0, . . . k, (k + 2)− µ2 · k, k̃ + 2, . . . , 2̃n, µ)
= Pf(0, 0̃, . . . k̃, (k + 2)− µ2 · k, k̃ + 2, . . . , 2̃n, µ).
Noticing (k + 2)− µ2 · k = k̃ + 1− µ · k̃, we get
Pf(0, . . . , k̂ + 1, . . . , 2n+ 1, µ) = Pf(0, 0̃, . . . ,
̂̃
k, . . . , 2̃n, µ)− µPf(0, 0̃, . . . ,
̂̃
k + 1, . . . , 2̃n, µ)
Discrete Spectral Transformations of SOPs and Associated DIS 11
= Ts[Pf(0, . . . , k̂, . . . , 2n− 1)]− µTs[Pf(0, . . . , k̂ + 1, . . . , 2n− 1)],
where we have used the relation (26). Therefore with the symmetry µ↔ λ, we have
Ts[Pf(0, 1, . . . , 2n− 2, 2n)] = Pf(0, 1, . . . , 2n− 1, 2n+ 1, µ)− µτ s+1,t
n ,
Tt[Pf(0, 1, . . . , 2n− 2, 2n)] = Pf(0, 1, . . . , 2n− 1, 2n+ 1, µ)− λτ s,t+1
n .
In a similar manner, we can also obtain
Ts ◦ Tt[Pf(0, 1, . . . , 2n− 2, 2n)] = λ−1µ Pf(0, 1, . . . , 2n− 1, 2n+ 1, µ)− (µ+ λ)τ s+1,t+1
n ,
Ts[Pf(0, 1, . . . , 2n− 1, 2n+ 1, z)] = z−1µ Pf(0, 1, . . . , 2n, 2n+ 2, µ, z)− µτ̂ s+1,t
n ,
Tt[Pf(0, 1, . . . , 2n− 1, 2n+ 1, z)] = z−1λ Pf(0, 1, . . . , 2n, 2n+ 2, λ, z)− λτ̂ s,t+1
n ,
Ts ◦ Tt[Pf(0, 1, . . . , 2n− 1, 2n+ 1, z)]
= z−1µ,λPf(0, 1, . . . , 2n+ 1, 2n+ 3, µ, λ, z)− (µ+ λ)τ̂ s+1,t+1
n .
These results immediately lead us to the conclusion. �
By virtue of Lemma 2, we can find the relation between τ s,tn , τ̂ s,tn , σs,tn and σ̂s,tn .
Theorem 5. The function τ s,tn , τ̂ s,tn , σs,tn and σ̂s,tn defined in (22) and (36) satisfy the following
four relations:
zµ,λλµσ
s,t
n+1τ̂
s+1,t+1
n−1 = zλτ
s+1,t
n σ̂s,t+1
n − zµτ s,t+1
n σ̂s+1,t
n + λµσ
s+1,t+1
n τ̂ s,tn ,
zµ,λλµτ
s,t
n+1σ̂
s+1,t+1
n−1 = zλσ
s+1,t
n τ̂ s,t+1
n − zµσs,t+1
n τ̂ s+1,t
n + λµτ
s+1,t+1
n σ̂s,tn ,
zµ,λλµσ
s,t
n+1τ̂
s+1,t+1
n = zλτ
s+1,t
n+1 σ̂s,t+1
n − zµτ s,t+1
n+1 σ̂s+1,t
n + λµσ
s+1,t+1
n τ̂ s,tn+1,
zµ,λλµτ
s,t
n+1σ̂
s+1,t+1
n = zλσ
s+1,t
n+1 τ̂
s,t+1
n − zµσs,t+1
n+1 τ̂
s+1,t
n + λµτ
s+1,t+1
n σ̂s,tn+1. (37)
Proof. Pfaffian identities (31), (32) and Lemma 2 directly lead us to the (37). This completes
the proof. �
If we take σ = τ , one finds that the equations (37) go to the discrete coupled KP equation (33).
Therefore, these equations can be considered as the natural extension of the discrete coupled
KP equation from the point of view of the SOPs. Hereafter we will call (33) as the extended
discrete coupled KP (edcKP) equation.
The edcKP equation gives us the contiguous relation among {φs,tn (z)}:
Theorem 6. Let Φs,t
n (z) be the vector function given by
Φs,t
n (z) =
(
φs,t2n(z)
φs,t2n+1(z)
)
.
The vector functions {Φs,t
n (z)}∞n=0 satisfy the following two vector linear equations:
zλΦs,t+1
n (z)− zµΦs+1,t
n (z) = zµ,λA
s,t
n Φs+1,t+1
n−1 (z)−Bs,t
n Φs,t
n (z),
zµ,λΦs+1,t+1
n (z)− Φn+1(z) = zλC
s,t
n Φs,t+1
n (z)− zµDs,t
n Φs+1,t
n , (38)
where the coefficients As,t
n , Bs,t
n , Cs,t
n , Ds,t
n can be written explicitly in terms of Pfaffians:
As,t
n = (λ− µ)
0
τ s,tn+1τ
s+1,t+1
n−1
σs+1,t
n σs,t+1
n
σs,tn+1σ
s+1,t+1
n−1
τ s+1,t
n τ s,t+1
n
0
,
12 H. Miki, H. Goda and S. Tsujimoto
Bs,t
n = (λ− µ)
0
τ s,tn τ s+1,t+1
n
σs+1,t
n σs,t+1
n
σs,tn σ
s+1,t+1
n
τ s+1,t
n τ s,t+1
n
0
,
Cs,t
n = (λ− µ)−1
0
τ s+1,t
n+1 τ s,t+1
n
σs,tn+1σ
s+1,t+1
n
σs+1,t
n+1 σ
s,t+1
n
τ s,tn+1τ
s+1,t+1
n
0
,
Ds,t
n = (λ− µ)−1
0
τ s,t+1
n+1 τ s+1,t
n
σs,tn+1σ
s+1,t+1
n
σs,t+1
n+1 σ
s+1,t
n
τ s,tn+1τ
s+1,t+1
n
0
. (39)
Proof. The definitions of {φs,tn }∞n=0 (35) and the edcKP equation (37) directly lead us to the
conclusion. This completes the proof. �
We can regard (38) as the discrete vector Lax pair of SOPs of both even and odd degree.
Therefore the discrete Lax pair shows us the following systems associated with SOPs of both
even and odd degree, which can be regarded as a discrete integrable system along with (34):
Theorem 7. From the compatibility condition of the discrete vector Lax pair (38), we can derive
the nonlinear system in 2 + 1 dimension which takes the 2× 2 matrix form:
As+1,t+1
n −As,t
n+1 + Bs,t
n+1 −Bs+1,t+1
n = Cs,t+1
n −Cs+1,t
n + Ds+1,t
n −Ds,t+1
n ,
As+1,t
n Cs+1,t
n−1 = As,t
n Cs+1,t
n , As,t+1
n Ds,t+1
n−1 = As,t
n Ds,t
n ,
Bs+1,t
n Ds+1,t
n = Bs,t
n+1D
s,t
n , Bs,t+1
n Ds,t+1
n = Bs,t
n+1D
s,t
n . (40)
Special solutions of the nonlinear system (40) can be written in terms of Pfaffians given by (39).
One can easily find that this system is the 2 × 2 matrix extension of the system (40) and
this is the nonlinear form of the edcKP equation. This system is clearly connected to the SOPs
of both even and odd degree and from its construction, we can regard the system (40) as the
discrete coutnerpart of the Pfaff lattice in (2 + 1)-dimensional form.
5 Concluding remarks
In this paper, we proposed the discrete spectral transformation of SOPs. From the discrete
spectral transformation of SOPs, we derived the discrete integrable systems both in (1 + 1)-
dimensional form and in (2 + 1)-dimensional form. In (1 + 1)-dimensional case, the system gives
the integrable discretization of the Pfaff lattice, though the system is of course not local. In
(2+1)-dimensional case, the system is local and corresponds only to the even-degree SOPs. This
system is also shown to be connected to discrete coupled KP equation. The discrete coupled
KP equation is known to be equivalent to the discretized Hirota–Ohta equation [11] in the
bilinear form levels. Recently, the discrete integrable system is proposed from the auxiliary
linear problems for the Hirota–Ohta (or DKP) system [4, 24].
We further extended the discrete integrable system associated with even-degree SOPs to
the system associated with both even and odd degree SOPs. This system can be described in
2 × 2 matrix form, which is related to an extension of the discrete coupled KP equation. This
Discrete Spectral Transformations of SOPs and Associated DIS 13
extension has never been considered before, although it is quite natural from the aspects of
SOPs.
Finally, let us mention for the relationship between our results and the random matrix theory.
As a by-product of the discrete transformation of SOPs, we found the compact factorization of
the kernel function of the correlation function for the orthogonal or symplectic ensembles in
terms of the two even-degree SOPs with respect to the different weights. This factorization is
not known as far as the authors know and indeed might be a new one. Moreover, it is known
that a matrix integral over orthogonal and symplectic ensembles of Hermitian matrices can be
considered as a τ -function of the coupled KP hierarchy [12], while our discrete integrable sys-
tems (34) and (40) are directly connected to discrete coupled KP and its extension, respectively.
We hope these results will bring a deeper understanding of the random matrix theory and several
applications in its works in the near future.
Acknowledgement
The authors would like to thank the Centre de recherches mathématiques (CRM) for its hos-
pitality. The work of H.M. is partially supported by a Grant-in-Aid for Japan Society for the
Promotion of Science (JSPS) Fellows. The research of S.T. is supported in part by KAK-
ENHI (22540224). The authors should like to thank the referees for the careful reading of the
manuscript and a lot of helpful suggestions and comments.
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1 Introduction
2 Skew orthogonal polynomials
3 Discrete integrable systems of SOPs in 2+1 dimension
4 An extension to 22 matrix form
5 Concluding remarks
References
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