Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice
The spectral transformation technique for symmetric RII polynomials is developed. Use of this technique reveals that the nonautonomous discrete modified KdV (nd-mKdV) lattice is directly connected with the RII chain. Hankel determinant solutions to the semi-infinite nd-mKdV lattice are also presente...
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Інститут математики НАН України
2013
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice / K. Maeda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 38 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1493652019-02-22T01:23:27Z Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice Maeda, K. Tsujimoto, S. The spectral transformation technique for symmetric RII polynomials is developed. Use of this technique reveals that the nonautonomous discrete modified KdV (nd-mKdV) lattice is directly connected with the RII chain. Hankel determinant solutions to the semi-infinite nd-mKdV lattice are also presented. 2013 Article Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice / K. Maeda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 38 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K35; 37K60; 42C05 DOI: http://dx.doi.org/10.3842/SIGMA.2013.073 http://dspace.nbuv.gov.ua/handle/123456789/149365 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The spectral transformation technique for symmetric RII polynomials is developed. Use of this technique reveals that the nonautonomous discrete modified KdV (nd-mKdV) lattice is directly connected with the RII chain. Hankel determinant solutions to the semi-infinite nd-mKdV lattice are also presented. |
| format |
Article |
| author |
Maeda, K. Tsujimoto, S. |
| spellingShingle |
Maeda, K. Tsujimoto, S. Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Maeda, K. Tsujimoto, S. |
| author_sort |
Maeda, K. |
| title |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice |
| title_short |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice |
| title_full |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice |
| title_fullStr |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice |
| title_full_unstemmed |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice |
| title_sort |
direct connection between the rii chain and the nonautonomous discrete modified kdv lattice |
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Інститут математики НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/149365 |
| citation_txt |
Direct Connection between the RII Chain and the Nonautonomous Discrete Modified KdV Lattice / K. Maeda, S. Tsujimoto // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 38 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT maedak directconnectionbetweentheriichainandthenonautonomousdiscretemodifiedkdvlattice AT tsujimotos directconnectionbetweentheriichainandthenonautonomousdiscretemodifiedkdvlattice |
| first_indexed |
2025-07-12T21:57:16Z |
| last_indexed |
2025-07-12T21:57:16Z |
| _version_ |
1837479957455962112 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 073, 12 pages
Direct Connection between the RII Chain
and the Nonautonomous Discrete
Modified KdV Lattice
Kazuki MAEDA and Satoshi TSUJIMOTO
Department of Applied Mathematics and Physics, Graduate School of Informatics,
Kyoto University, Kyoto 606-8501, Japan
E-mail: kmaeda@amp.i.kyoto-u.ac.jp, tujimoto@i.kyoto-u.ac.jp
Received September 20, 2013, in final form November 22, 2013; Published online November 26, 2013
http://dx.doi.org/10.3842/SIGMA.2013.073
Abstract. The spectral transformation technique for symmetric RII polynomials is de-
veloped. Use of this technique reveals that the nonautonomous discrete modified KdV
(nd-mKdV) lattice is directly connected with the RII chain. Hankel determinant solutions
to the semi-infinite nd-mKdV lattice are also presented.
Key words: orthogonal polynomials; spectral transformation; RII chain; nonautonomous
discrete modified KdV lattice
2010 Mathematics Subject Classification: 37K35; 37K60; 42C05
1 Introduction
In the theory of integrable systems, orthogonal polynomials play an important role. In particular,
the spectral transformation technique yields various integrable systems and particular solu-
tions [24, 26, 27]. The spectral transformation for orthogonal polynomials is a mapping from an
orthogonal polynomial sequence to another orthogonal polynomial sequence. We can view the
three-term recurrence relation and the spectral transformation for orthogonal polynomials as
a Lax pair, where the compatibility condition induces an integrable system. Furthermore, the
determinant structure of orthogonal polynomials allows us to derive particular solutions to the
associated integrable system. During the last fifteen years, many researchers have extended this
technique to generalized (bi)orthogonal functions and have exploited novel integrable systems
that have rich properties [1, 2, 3, 12, 14, 15, 16, 17, 33, 34, 35, 37]. In this paper, we will
extend the spectral transformation technique for symmetric orthogonal polynomials and the
associated discrete integrable system to symmetric RII polynomials. Our motivation comes
from applications of discrete integrable systems to numerical algorithms. It is well-known that
the discrete integrable system associated with orthogonal polynomials is the nonautonomous
discrete Toda (nd-Toda) lattice and that its time evolution equation is the same as the recurrence
relation of the dqds algorithm [5], a fast and accurate eigenvalue or singular value algorithm.
Similarly, the discrete integrable system associated with symmetric orthogonal polynomials is the
nonautonomous discrete Lotka–Volterra (nd-LV) lattice, which can compute singular values [9].
By using the spectral transformation technique, we can easily derive a direct connection between
the nd-Toda lattice and the nd-LV lattice. This connection was used to develop the mdLVs
algorithm, which is an improved version of the singular value algorithm based on the nd-LV
lattice [10].
Recently, the authors have been developing a generalized eigenvalue algorithm based on
the RII chain [13]. Since the RII chain is associated with RII polynomials, a generalization
of orthogonal polynomials from the point of view of Padé approximation or the eigenvalue
mailto:kmaeda@amp.i.kyoto-u.ac.jp
mailto:tujimoto@i.kyoto-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2013.073
2 K. Maeda and S. Tsujimoto
problem [8, 18, 28], the proposed algorithm has good properties similar to the dqds algorithm.
These studies motivate us to find the discrete integrable system associated with symmetric RII
polynomials and its connection with the RII chain. The derived discrete integrable system may
become the basis for developing good numerical algorithms for generalized eigenvalue problems
or Padé approximations, for example.
This paper is organized as follows. In Section 2, we briefly recall the derivation of the nd-
Toda lattice and the nd-LV lattice from the theory of spectral transformations for ordinary and
symmetric orthogonal polynomials, respectively. We also review the direct connection (Miura
transformation) between the nd-Toda lattice and the nd-LV lattice. In Section 3, we extend the
framework presented in Section 2 to RII polynomials. We then demonstrate that the spectral
transformations for symmetric RII polynomials give rise to the nonautonomous discrete modified
KdV (nd-mKdV) lattice. Particular solutions to the semi-infinite nd-mKdV lattice and the
direct connection between the RII chain and the nd-mKdV lattice are also derived. Section 4 is
devoted to concluding remarks.
2 Derivation of the nd-Toda lattice and the nd-LV lattice
2.1 Orthogonal polynomials and the nd-Toda lattice
Monic orthogonal polynomials are defined by a three-term recurrence relation in the form
φk,t−1(x) := 0, φk,t0 (x) := 1,
φk,tn+1(x) :=
(
x− ak,tn
)
φk,tn (x)− bk,tn φk,tn−1(x), n = 0, 1, 2, . . . , (1)
where ak,tn ∈ R, bk,tn ∈ R \ {0}, and k, t ∈ Z indicate discrete time. By definition, φk,tn (x)
is a monic polynomial of degree n. If some constant hk,t0 ∈ R \ {0} is fixed, then Favard’s
theorem [4] provides a unique linear functional Lk,t : R[x] → R such that the orthogonality
relation
Lk,t[xmφk,tn (x)] = hk,tn δm,n, n = 0, 1, 2, . . . , m = 0, 1, . . . , n, (2)
holds, where
hk,tn = hk,t0 bk,t1 bk,t2 · · · bk,tn , n = 1, 2, 3, . . . ,
and δm,n is the Kronecker delta.
Let us introduce time evolution into the orthogonal polynomials through spectral transfor-
mations. First, the spectral transformations for the k-direction are
xφk+1,t
n (x) = φk,tn+1(x) + qk,tn φk,tn (x), (3a)
φk,tn (x) = φk+1,t
n (x) + ek,tn φk+1,t
n−1 (x), (3b)
where
qk,tn := −φ
k,t
n+1(0)
φk,tn (0)
, ek,tn :=
Lk,t[xnφk,tn (x)]
Lk+1,t[xn−1φk+1,t
n−1 (x)]
, (4)
Lk+1,t[π(x)] := Lk,t[xπ(x)] for all π(x) ∈ R[x]. (5)
It is readily verified that
{
φk+1,t
n (x)
}∞
n=0
are monic orthogonal polynomials with respect to the
linear functional Lk+1,t. Similarly, the spectral transformations for the t-direction are(
x+ s(t)
)
φk,t+1
n (x) = φk,tn+1(x) + q̃k,tn φk,tn (x), (6a)
φk,tn (x) = φk,t+1
n (x) + ẽk,tn φk,t+1
n−1 (x), (6b)
Direct Connection between the RII Chain and the ND-MKdV Lattice 3
where s(t) is a nonzero parameter depending on t and
q̃k,tn := −φ
k,t
n+1(−s(t))
φk,tn (−s(t))
, ẽk,tn :=
Lk,t
[
xnφk,tn (x)
]
Lk,t+1
[
xn−1φk,t+1
n−1 (x)
] , (7)
Lk,t+1[π(x)] := Lk,t
[(
x+ s(t)
)
π(x)
]
for all π(x) ∈ R[x]. (8)
The only difference between the transformations for the k-direction (3) and the t-direction (6)
is the parameter s(t). Fig. 1 illustrates the relations among the monic orthogonal polynomials,
the spectral transformations and the dependent variables.
{φk,tn (x)}∞n=0 {φk,t+1
n (x)}∞n=0{φk,t−1n (x)}∞n=0
{φk+1,t
n (x)}∞n=0 {φk+1,t+1
n (x)}∞n=0{φk+1,t−1
n (x)}∞n=0
{φk−1,tn (x)}∞n=0 {φk−1,t+1
n (x)}∞n=0{φk−1,t−1n (x)}∞n=0
{qk,tn }∞n=0 {ek,tn }∞n=1 {qk,t+1
n }∞n=0 {ek,t+1
n }∞n=1{qk,t−1n }∞n=0 {ek,t−1n }∞n=1
{qk−1,tn }∞n=0 {ek−1,tn }∞n=1 {qk−1,t+1
n }∞n=0 {ek−1,t+1
n }∞n=1{qk−1,t−1n }∞n=0 {ek−1,t−1n }∞n=1
{q̃k,tn }∞n=0, s
(t)
{ẽk,tn }∞n=1
{q̃k+1,t
n }∞n=0, s
(t)
{ẽk+1,t
n }∞n=1
{q̃k−1,tn }∞n=0, s
(t)
{ẽk−1,tn }∞n=1
{q̃k,t−1n }∞n=0, s
(t−1)
{ẽk,t−1n }∞n=1
{q̃k+1,t−1
n }∞n=0, s
(t−1)
{ẽk+1,t−1
n }∞n=1
{q̃k−1,t−1n }∞n=0, s
(t−1)
{ẽk−1,t−1n }∞n=1
Figure 1. Chain of the spectral transformations for monic orthogonal polynomials.
Relations (1), (3) and (6) yield
φk,tn+1(x) =
(
x− ak,tn
)
φk,tn (x)− bk,tn φk,tn−1(x)
=
(
x−
(
qk,tn + ek,tn
))
φk,tn (x)− qk,tn−1e
k,t
n φk,tn−1(x)
=
(
x−
(
qk−1,t
n + ek−1,t
n+1
))
φk,tn (x)− qk−1,t
n ek−1,t
n φk,tn−1(x)
=
(
x−
(
q̃k,tn + ẽk,tn − s(t)
))
φk,tn (x)− q̃k,tn−1ẽ
k,t
n φk,tn−1(x)
=
(
x−
(
q̃k,t−1
n + ẽk,t−1
n+1 − s(t−1)
))
φk,tn (x)− q̃k,t−1
n ẽk,t−1
n φk,tn−1(x).
Hence, for consistency, the compatibility conditions
ak,tn = qk,tn + ek,tn = qk−1,t
n + ek−1,t
n+1 = q̃k,tn + ẽk,tn − s(t) = q̃k,t−1
n + ẽk,t−1
n+1 − s(t−1), (9a)
bk,tn = qk,tn−1e
k,t
n = qk−1,t
n ek−1,t
n = q̃k,tn−1ẽ
k,t
n = q̃k,t−1
n ẽk,t−1
n , (9b)
ek,t0 = ẽk,t0 = 0 for all k and t, (9c)
must be satisfied. These are the time evolution equations of the semi-infinite nd-Toda lattice.
Equations (9) give the relations among the recurrence coefficients of
{
φk,tn (x)
}∞
n=0
and the
dependent variables around
{
φk,tn (x)
}∞
n=0
in the diagram (Fig. 1).
Define the moment of the linear functional L0,t by
µ(t)
m := L0,t
[
xm
]
.
Note that (5) gives the relation
Lk,t
[
xm
]
= L0,t
[
xk+m
]
= µ
(t)
k+m.
4 K. Maeda and S. Tsujimoto
Further, (8) gives the dispersion relation
µ(t+1)
m = µ
(t)
m+1 + s(t)µ(t)
m . (10)
We should remark that, if a concrete representation of the initial linear functional L0,0 is given
by a weighted integral, then the moment may be represented concretely as
µ(t)
m =
∫
Ω
w(x)xm
t−1∏
t′=0
(
x+ s(t′)
)
dx,
where Ω is some interval on the real line and w(x) is a weight function defined on Ω.
The determinant expression of the monic orthogonal polynomials
{
φk,tn (x)
}∞
n=0
is given by
φk,tn (x) =
1
τk,tn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
µ
(t)
k µ
(t)
k+1 · · · µ
(t)
k+n−1 µ
(t)
k+n
µ
(t)
k+1 µ
(t)
k+2 · · · µ
(t)
k+n µ
(t)
k+n+1
...
...
...
...
µ
(t)
k+n−1 µ
(t)
k+n · · · µ
(t)
k+2n−2 µ
(t)
k+2n−1
1 x · · · xn−1 xn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
, (11)
where τk,tn is the Hankel determinant of order n:
τk,t−1 := 0, τk,t0 := 1, τk,tn :=
∣∣µ(t)
k+i+j
∣∣
0≤i,j≤n−1
, n = 1, 2, 3, . . . .
One can readily verify that the right hand side of (11) is a monic polynomial of degree n and
satisfies the orthogonality relation (2). This determinant expression (11) and the dispersion
relation (10) enable us to give Hankel determinant solutions to the nd-Toda lattice (9); from (4)
and (7), we obtain
qk,tn =
τk,tn τk+1,t
n+1
τk,tn+1τ
k+1,t
n
, ek,tn =
τk,tn+1τ
k+1,t
n−1
τk,tn τk+1,t
n
, (12a)
q̃k,tn =
τk,tn τk,t+1
n+1
τk,tn+1τ
k,t+1
n
, ẽk,tn =
τk,tn+1τ
k,t+1
n−1
τk,tn τk,t+1
n
. (12b)
2.2 Symmetric orthogonal polynomials and the nd-LV lattice
Next, we consider the polynomial sequence
{
σk,tn (x)
}∞
n=0
defined by
σk,t2n+i(x) := xiφk+i,t
n
(
x2
)
, n = 0, 1, 2, . . . , i = 0, 1.
By definition, σk,tn (x) is a monic polynomial of degree n and has the symmetry property
σk,tn (−x) = (−1)nσk,tn (x).
Further, {σk,tn (x)}∞n=0 are orthogonal with respect to the linear functional Sk,t defined by
Sk,t
[
x2m
]
:= Lk,t
[
xm
]
, Sk,t
[
x2m+1
]
:= 0, m = 0, 1, 2, . . . .{
σk,tn (x)
}∞
n=0
are called monic symmetric orthogonal polynomials.
Direct Connection between the RII Chain and the ND-MKdV Lattice 5
From (3), we have the relations
x2φk+1,t
n
(
x2
)
= φk,tn+1
(
x2
)
+ qk,tn φk,tn
(
x2
)
,
xφk,tn
(
x2
)
= xφk+1,t
n
(
x2
)
+ ek,tn xφk+1,t
n−1
(
x2
)
.
These relations lead us to the three-term recurrence relation that {σk,tn (x)}∞n=0 satisfy:
σk,t2n+2(x) = xσk,t2n+1(x)− qk,tn σk,t2n (x), (13a)
σk,t2n+1(x) = xσk,t2n (x)− ek,tn σk,t2n−1(x). (13b)
Spectral transformations for {σk,tn (x)}∞n=0 are also induced from (6):(
x2 + s(t)
)
σk,t+1
2n (x) = σk,t2n+2(x) + q̃k,tn σk,t2n (x), (14a)(
x2 + s(t)
)
σk,t+1
2n+1 (x) = σk,t2n+3(x) + q̃k+1,t
n σk,t2n+1(x), (14b)
σk,t2n (x) = σk,t+1
2n (x) + ẽk,tn σk,t+1
2n−2 (x), (14c)
σk,t2n+1(x) = σk,t+1
2n+1 (x) + ẽk+1,t
n σk,t+1
2n−1 (x). (14d)
Relations (13) and (14) show that there exist variables vk,tn satisfying the relations(
x2 + s(t)
)
σk,t+1
n (x) = xσk,tn+1(x) +
(
s(t) + vk,tn
)
σk,tn (x), (15a)(
s(t) + vk,tn
)
σk,tn (x) = s(t)σk,t+1
n (x) + vk,tn xσk,t+1
n−1 (x). (15b)
Relations (15) yield
σk,tn+1(x) = xσk,tn (x)− vk,tn
(
1 +
(
s(t)
)−1
vk,tn−1
)
σk,tn−1(x)
= xσk,tn (x)− vk,t−1
n
(
1 +
(
s(t−1)
)−1
vk,t−1
n+1
)
σk,tn−1(x). (16)
Hence, the compatibility condition
vk,tn
(
1 +
(
s(t)
)−1
vk,tn−1
)
= vk,t−1
n
(
1 +
(
s(t−1)
)−1
vk,t−1
n+1
)
, (17a)
vk,t0 = 0 for all k and t, (17b)
must be satisfied. This is the time evolution equation of the semi-infinite nd-LV lattice.
From relations (13)–(16), we obtain the Miura transformation between the nd-Toda lattice (9)
and the nd-LV lattice (17):
qk,tn = vk,t2n+1
(
1 +
(
s(t)
)−1
vk,t2n
)
= vk,t−1
2n+1
(
1 +
(
s(t−1)
)−1
vk,t−1
2n+2
)
,
ek,tn = vk,t2n
(
1 +
(
s(t)
)−1
vk,t2n−1
)
= vk,t−1
2n
(
1 +
(
s(t−1)
)−1
vk,t−1
2n+1
)
,
q̃k,tn = s(t)
(
1 +
(
s(t)
)−1
vk,t2n+1
)(
1 +
(
s(t)
)−1
vk,t2n
)
= s(t)
(
1 +
(
s(t)
)−1
vk−1,t
2n+1
)(
1 +
(
s(t)
)−1
vk−1,t
2n+2
)
,
ẽk,tn =
(
s(t)
)−1
vk,t2nv
k,t
2n−1 =
(
s(t)
)−1
vk−1,t
2n vk−1,t
2n+1 .
In addition, from relations (15), we obtain
1 +
(
s(t)
)−1
vk,t2n =
σk,t+1
2n (0)
σk,t2n (0)
=
φk,t+1
n (0)
φk,tn (0)
,
1 +
(
s(t)
)−1
vk,t2n−1 =
(
−s(t)
)−1/2 σk,t2n
(
(−s(t))1/2
)
σk,t2n−1
(
(−s(t))1/2
) = −
(
s(t)
)−1 φk,tn (−s(t))
φk+1,t
n−1 (−s(t))
.
6 K. Maeda and S. Tsujimoto
By using these relations and the solutions to the nd-Toda lattice (12), we obtain Hankel deter-
minant solutions to the nd-LV lattice (17):
vk,t2n+1 = qk,tn
φk,tn (0)
φk,t+1
n (0)
=
τk,t+1
n τk+1,t
n+1
τk,tn+1τ
k+1,t+1
n
,
vk,t2n = −s(t)ek,tn
φk+1,t
n−1 (−s(t))
φk,tn (−s(t))
= s(t) τ
k,t
n+1τ
k+1,t+1
n−1
τk,t+1
n τk+1,t
n
.
3 Derivation of the RII chain and the nd-mKdV lattice
We will apply the framework constructed in the previous section to monic RII polynomials and
derive the nd-mKdV lattice.
3.1 RII polynomials and the RII chain
Monic RII polynomials are defined by the three-term recurrence relation of the form
ϕk,t−1(x) := 0, ϕk,t0 (x) := 1,
ϕk,tn+1(x) :=
((
1 + βk,tn
)
x− αk,tn
)
ϕk,tn (x)− βk,tn (x+ γk+t+2n−2)(x+ γk+t+2n−1)ϕk,tn−1(x),
n = 0, 1, 2, . . . ,
where αk,tn ∈ R and βk,tn , γk+t+n ∈ R \ {0}. If nonzero constants hk,t0 and hk,t1 are fixed, then
a Favard-type theorem [8] guarantees the existence of a unique linear functional Lk,t such that
the orthogonality relation
Lk,t
[
xmϕk,tn (x)∏2n−1
j=0 (x+ γk+t+j)
]
= hk,tn δm,n, n = 0, 1, 2, . . . , m = 0, 1, . . . , n,
holds, where hk,tn , n = 2, 3, . . . are nonzero constants. Note that Lk,t is defined on the vector
space spanned by 1∏l−1
j=0(x+γk+t+j)
, l = 0, 1, 2, . . . .
As in the case of monic orthogonal polynomials, we introduce the time evolution of the monic
RII polynomials by the following spectral transformations:(
1 + qk,tn
)
xϕk+1,t
n (x) = ϕk,tn+1(x) + qk,tn (x+ γk+t+2n)ϕk,tn (x), (18a)(
1 + ek,tn
)
ϕk,tn (x) = ϕk+1,t
n (x) + ek,tn (x+ γk+t+2n−1)ϕk+1,t
n−1 (x), (18b)(
1 + q̃k,tn
)(
x+ s(t)
)
ϕk,t+1
n (x) = ϕk,tn+1(x) + q̃k,tn (x+ γk+t+2n)ϕk,tn (x), (18c)(
1 + ẽk,tn
)
ϕk,tn (x) = ϕk,t+1
n (x) + ẽk,tn (x+ γk+t+2n−1)ϕk,t+1
n−1 (x). (18d)
These spectral transformations were originally introduced by Zhedanov [38]. By choosing the
variables qk,tn , ek,tn , q̃k,tn and ẽk,tn as above, the leading coefficients of both sides of (18) become
equal. The time evolution of the linear functional is also given by
Lk+1,t[ρ(x)] := Lk,t
[
x
x+ γk+t
ρ(x)
]
, Lk,t+1[ρ(x)] := Lk,t
[
x+ s(t)
x+ γk+t
ρ(x)
]
(19)
for rational functions ρ(x). One can verify that
{
ϕk+1,t
n (x)
}∞
n=0
and
{
ϕk,t+1
n (x)
}∞
n=0
are both also
monic RII polynomials. The spectral transformations (18) induce the time evolution equations
Direct Connection between the RII Chain and the ND-MKdV Lattice 7
of the semi-infinite monic-type RII chain:
αk,tn = γk+t+2nq
k,t
n + γk+t+2n−1e
k,t
n
1 + qk,tn
1 + qk,tn−1
= γk+t+2n−1q
k−1,t
n
1 + ek−1,t
n+1
1 + ek−1,t
n
+ γk+t+2ne
k−1,t
n+1
= γk+t+2nq̃
k,t
n + γk+t+2n−1ẽ
k,t
n
1 + q̃k,tn
1 + q̃k,tn−1
− s(t)
(
1 + q̃k,tn
)(
1 + ẽk,tn
)
= γk+t+2n−1q̃
k,t−1
n
1 + ẽk,t−1
n+1
1 + ẽk,t−1
n
+ γk+t+2nẽ
k,t−1
n+1 − s(t−1)
(
1 + q̃k,t−1
n
)(
1 + ẽk,t−1
n+1
)
, (20a)
βk,tn = qk,tn−1e
k,t
n
1 + qk,tn
1 + qk,tn−1
= qk−1,t
n ek−1,t
n
1 + ek−1,t
n+1
1 + ek−1,t
n
= q̃k,tn−1ẽ
k,t
n
1 + q̃k,tn
1 + q̃k,tn−1
= q̃k,t−1
n ẽk,t−1
n
1 + ẽk,t−1
n+1
1 + ẽk,t−1
n
, (20b)
ek,t0 = ẽk,t0 = 0 for all k and t. (20c)
Note that the RII chain was originally introduced by Spiridonov and Zhedanov [28]. The original
chain is described by three equations and four types of dependent variables with one constraint.
The monic-type version (20) is, however, described by essentially only two equations and two
types of dependent variables; since we are now considering two time variables k and t, there are
four types of dependent variables.
We define the moment of the linear functional L0,t by
µ
(t)
m,l := L0,t
[
xm∏l−1
j=0(x+ γt+j)
]
.
Note that the time evolution of the linear functional (19) gives the relation
Lk,t
[
xm∏l−1
j=0(x+ γk+t+j)
]
= L0,t
[
xk+m∏k+l−1
j=0 (x+ γt+j)
]
= µ
(t)
k+m,k+l
and the dispersion relations
µ
(t+1)
m,l = µ
(t)
m+1,l+1 + s(t)µ
(t)
m,l+1, µ
(t)
m,l = µ
(t)
m+1,l+1 + γt+lµ
(t)
m,l+1. (21)
Then, the determinant expression of the monic RII polynomials
{
ϕk,tn (x)
}∞
n=0
is given by
ϕk,tn (x) =
1
τk,2n,tn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
µ
(t)
k,k+2n µ
(t)
k+1,k+2n · · · µ
(t)
k+n−1,k+2n µ
(t)
k+n,k+2n
µ
(t)
k+1,k+2n µ
(t)
k+2,k+2n · · · µ
(t)
k+n,k+2n µ
(t)
k+n+1,k+2n
...
...
...
...
µ
(t)
k+n−1,k+2n µ
(t)
k+n,k+2n · · · µ
(t)
k+2n−2,k+2n µ
(t)
k+2n−1,k+2n
1 x · · · xn−1 xn
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
, (22)
where τk,l,tn is the Hankel determinant of order n:
τk,l,t−1 := 0, τk,l,t0 := 1, τk,l,tn :=
∣∣µ(t)
k+i+j,k+l
∣∣
0≤i,j≤n−1
, n = 1, 2, 3, . . . . (23)
8 K. Maeda and S. Tsujimoto
We should remark that the Casorati determinant representation of the RII polynomials was found
by Spiridonov and Zhedanov [31]. The Hankel determinant expression given above reflects
the structure of the discrete two-dimensional Toda hierarchy [34]. By using the determinant
expression (22), the dispersion relation (21), and a determinant identity called Plücker relation,
we can find the Hankel determinant solutions to the semi-infinite monic-type RII chain:
qk,tn = (γk+t+2n)−1 τk,2n,tn τk+1,2n+1,t
n+1
τk,2n+2,t
n+1 τk+1,2n−1,t
n
, ek,tn = γk+t+2n
τk,2n+1,t
n+1 τk+1,2n−2,t
n−1
τk,2n−1,t
n τk+1,2n,t
n
,
q̃k,tn =
(
γk+t+2n − s(t)
)−1 τk,2n,tn τk,2n+1,t+1
n+1
τk,2n+2,t
n+1 τk,2n−1,t+1
n
, ẽk,tn =
(
γk+t+2n − s(t)
)τk,2n+1,t
n+1 τk,2n−2,t+1
n−1
τk,2n−1,t
n τk,2n,t+1
n
.
3.2 Symmetric RII polynomials and the nd-mKdV lattice
We introduce a symmetric version of the monic RII polynomials, which is an analogue of the
monic symmetric orthogonal polynomials.
Let us define a polynomial sequence
{
ςk,tn (x)
}∞
n=0
by
ςk,t2n+i(x) := xiϕk+i,t
n
(
x2
)
, n = 0, 1, 2, . . . , i = 0, 1.
The corresponding linear functional Sk,t is given by
Sk,t
[
x2m∏l−1
j=0(x2 + γk+t+j)
]
:= Lk,t
[
xm∏l−1
j=0(x+ γk+t+j)
]
,
Sk,t
[
x2m+1∏l−1
j=0(x2 + γk+t+j)
]
:= 0, l = 0, 1, 2, . . . , m = 0, 1, . . . , l.
The spectral transformations (18) yield the three-term recurrence relations
ςk,t2n+2(x) =
(
1 + qk,tn
)
xςk,t2n+1(x)− qk,tn
(
x2 + γk+t+2n
)
ςk,t2n (x), (24a)
ςk,t2n+1(x) =
(
1 + ek,tn
)
xςk,t2n (x)− ek,tn
(
x2 + γk+t+2n−1
)
ςk,t2n−1(x), (24b)
and the spectral transformations for the monic symmetric RII polynomials
{
ςk,tn (x)
}∞
n=0
:
(
1 + q̃k,tn
)(
x2 + s(t)
)
ςk,t+1
2n (x) = ςk,t2n+2(x) + q̃k,tn
(
x2 + γk+t+2n
)
ςk,t2n (x), (25a)(
1 + q̃k+1,t
n
)(
x2 + s(t)
)
ςk,t+1
2n+1 (x) = ςk,t2n+3(x) + q̃k+1,t
n
(
x2 + γk+t+2n+1
)
ςk,t2n+1(x), (25b)(
1 + ẽk,tn
)
ςk,t2n (x) = ςk,t+1
2n (x) + ẽk,tn
(
x2 + γk+t+2n−1
)
ςk,t+1
2n−2 (x), (25c)(
1 + ẽk+1,t
n
)
ςk,t2n+1(x) = ςk,t+1
2n+1 (x) + ẽk+1,t
n
(
x2 + γk+t+2n
)
ςk,t+1
2n−1 (x). (25d)
Relations (24) and (25) show that there exist variables vk,tn satisfying the relations(
γk+t+n + vk,tn
)(
x2 + s(t)
)
ςk,t+1
n (x)
=
(
γk+t+n − s(t)
)
xςk,tn+1(x) +
(
s(t) + vk,tn
)(
x2 + γk+t+n
)
ςk,tn (x), (26a)
γk+t+n
(
s(t) + vk,tn
)
ςk,tn (x)
= s(t)(γk+t+n + vk,tn )ςk,t+1
n (x) +
(
γk+t+n − s(t)
)
vk,tn xςk,t+1
n−1 (x). (26b)
Direct Connection between the RII Chain and the ND-MKdV Lattice 9
Relations (26) yield
ςk,tn+1(x) =
(
1 + (γk+t+n−1)−1vk,tn
1 +
(
s(t)
)−1
vk,tn−1
1 + (γk+t+n−1)−1vk,tn−1
)
xςk,tn (x)
− (γk+t+n−1)−1vk,tn
1 +
(
s(t)
)−1
vk,tn−1
1 + (γk+t+n−1)−1vk,tn−1
(
x2 + γk+t+n−1
)
ςk,tn−1(x)
=
(
1 + (γk+t+n−1)−1vk,t−1
n
1 +
(
s(t−1)
)−1
vk,t−1
n+1
1 + (γk+t+n)−1vk,t−1
n+1
)
xςk,tn (x)
− (γk+t+n−1)−1vk,t−1
n
1 +
(
s(t−1)
)−1
vk,t−1
n+1
1 + (γk+t+n)−1vk,t−1
n+1
(
x2 + γk+t+n−1
)
ςk,tn−1(x).
Hence, the compatibility condition
vk,tn
1 +
(
s(t)
)−1
vk,tn−1
1 + (γk+t+n−1)−1vk,tn−1
= vk,t−1
n
1 +
(
s(t−1)
)−1
vk,t−1
n+1
1 + (γk+t+n)−1vk,t−1
n+1
, (27a)
vk,t0 = 0 for all k and t, (27b)
must be satisfied. This is the time evolution equation of the semi-infinite nd-mKdV lattice,
a nonautonomous version of the discrete mKdV lattice [32]. Note that the nd-mKdV lattice (27)
reduces to the nd-LV lattice (17) as γk+t+n →∞.
The Miura transformation between the monic-type RII chain (20) and the nd-mKdV lat-
tice (27) is obtained as follows:
qk,tn = (γk+t+2n)−1vk,t2n+1
1 +
(
s(t)
)−1
vk,t2n
1 + (γk+t+2n)−1vk,t2n
= (γk+t+2n)−1vk,t−1
2n+1
1 +
(
s(t−1)
)−1
vk,t−1
2n+2
1 + (γk+t+2n+1)−1vk,t−1
2n+2
,
ek,tn = (γk+t+2n−1)−1vk,t2n
1 +
(
s(t)
)−1
vk,t2n−1
1 + (γk+t+2n−1)−1vk,t2n−1
= (γk+t+2n−1)−1vk,t−1
2n
1 +
(
s(t−1)
)−1
vk,t−1
2n+1
1 + (γk+t+2n)−1vk,t−1
2n+1
,
q̃k,tn =
(
γk+t+2n − s(t)
)−1
s(t)
(
1 +
(
s(t)
)−1
vk,t2n+1
)(
1 +
(
s(t)
)−1
vk,t2n
)
=
(
γk+t+2n − s(t)
)−1
s(t)
(
1 +
(
s(t)
)−1
vk−1,t
2n+1
)(
1 +
(
s(t)
)−1
vk−1,t
2n+2
)
,
ẽk,tn =
(
γk+t+2n − s(t)
)
vk,t2nv
k,t
2n−1
s(t)γk+t+2n−1γk+t+2n
(
1 + (γk+t+2n)−1vk,t2n
)(
1 + (γk+t+2n−1)−1vk,t2n−1
)
=
(
γk+t+2n − s(t)
)
vk−1,t
2n vk−1,t
2n+1
s(t)γk+t+2n−1γk+t+2n
(
1 + (γk+t+2n−1)−1vk−1,t
2n
)(
1 + (γk+t+2n)−1vk−1,t
2n+1
) .
Furthermore, from relations (26), we obtain
1 +
(
s(t)
)−1
vk,t2n
1 + (γk+t+2n)−1vk,t2n
=
ϕk,t+1
n (0)
ϕk,tn (0)
,
1 +
(
s(t)
)−1
vk,t2n−1 = (−s(t))−1 ϕk,tn (−s(t))
ϕk+1,t
n−1 (−s(t))
,
1 + (γk+t+2n−1)−1vk,t2n−1 = (−γk+t+2n−1)−1 ϕk,tn (−γk+t+2n−1)
ϕk+1,t+1
n−1 (−γk+t+2n−1)
.
10 K. Maeda and S. Tsujimoto
Hence, Hankel determinant solutions to the nd-mKdV lattice (27) are given by
vk,t2n+1 = γk+t+2nq
k,t
n
ϕk,tn (0)
ϕk,t+1
n (0)
=
τk,2n,t+1
n τk+1,2n+1,t
n+1
τk,2n+2,t
n+1 τk+1,2n−1,t+1
n
,
vk,t2n = s(t)ek,tn
ϕk+1,t
n−1 (−s(t))
ϕk,tn (−s(t))
ϕk,tn (−γk+t+2n−1)
ϕk+1,t+1
n−1 (−γk+t+2n−1)
= s(t)γk+t+2n
τk,2n+1,t
n+1 τk+1,2n−2,t+1
n−1
τk,2n−1,t+1
n τk+1,2n,t
n
.
Remark 1. Spiridonov [29] first considered spectral transformations for the (not monic) sym-
metric RII polynomials. By using spectral transformations, he derived a generalization of the
nd-LV lattice (17), which is more complicated than the nd-mKdV lattice (27). We have con-
sidered the monic symmetric RII polynomials and their spectral transformations which possess
the following symmetry. Consider an independent variable transformation t′ = −k − t− n and
introduce ς̃k,t
′
n (x) := ςk,−k−t
′−n+1
n (x), ṽk,t
′
n := vk,−k−t
′−n
n , s̃k+t′+n := s(−k−t′−n) and γ̃(t′) := γ−t′ .
Then, the spectral transformations for the monic symmetric RII polynomials (26) may be rewrit-
ten as(
s̃k+t′+n + ṽk,t
′
n
)(
x2 + γ̃(t′)
)
ς̃k,t
′+1
n (x)
=
(
s̃k+t′+n − γ̃(t′)
)
xς̃k,t
′
n+1(x) +
(
γ̃(t′) + ṽk,t
′
n
)(
x2 + s̃k+t′+n
)
ς̃k,t
′
n (x),
s̃k+t′+n
(
γ̃(t′) + ṽk,t
′
n
)
ς̃k,t
′
n (x)
= γ̃(t′)
(
s̃k+t′+n + ṽk,t
′
n
)
ς̃k,t
′+1
n (x) +
(
s̃k+t′+n − γ̃(t′)
)
ṽk,t
′
n xς̃k,t
′+1
n−1 (x),
so that the roles of the parameters are replaced. Using the symmetric form of the spectral
transformations (26), we can derive the corresponding discrete integrable system in a simpler
form.
In another study, Spiridonov et al. [30] derived a discrete integrable system called the FST
chain and discussed its connection to the RII chain. The time evolution equation of the FST
chain is
γk+t+n − s(t) +Ak,tn Ak,tn−1
Ak,tn
=
γk+t+n−1 − s(t−1) +Ak,t−1
n Ak,t−1
n+1
Ak,t−1
n
,
Ak,t−1 = 0 for all k and t.
Particular solutions to the FST chain may also be expressed by the Hankel determinant (23):
Ak,t2n =
(
γk+t+2n − s(t)
)τk,2n+1,t
n+1 τk,2n−1,t+1
n
τk,2n,tn τk,2n,t+1
n
,
Ak,t2n+1 =
(
γk+t+2n+1 − s(t)
) τk,2n+2,t
n+1 τk,2n,t+1
n
τk,2n+1,t
n+1 τk,2n+1,t+1
n+1
.
Similarly, we have the discrete potential KdV lattice(
εk,t−1
n − εk,tn
)(
εk,t−1
n+1 − εk,tn−1
)
= γk+t+n−1 − s(t−1),
εk,t−1 = 0 for all k and t,
and its Hankel determinant solutions
εk,t2n =
τk,2n,tn+1
τk,2n,tn
, εk,t2n+1 =
τk,2n+1,t
n
τk,2n+1,t
n+1
.
Therefore, these systems and the nd-mKdV lattice (27) are connected via the bilinear formalism.
Direct Connection between the RII Chain and the ND-MKdV Lattice 11
4 Concluding remarks
In this paper, we developed the spectral transformation technique for symmetric RII polynomials
and derived the nd-mKdV lattice as the compatibility condition. Moreover, we obtained a direct
connection between the RII chain and the nd-mKdV lattice. It is easily verified by numerical
experiments that the obtained nd-mKdV lattice with a non-periodic finite lattice condition
can compute the generalized eigenvalues of the tridiagonal matrix pencil that corresponds to
the RII polynomials through the Miura transformation. More practical applications of the nd-
mKdV lattice to numerical algorithms are left for future work. In particular, the application to
generalized singular value decomposition [36] will be discussed in detail.
In recent studies, various discrete Painlevé equations have been obtained as reductions of
discrete integrable systems [6, 7, 20, 21, 22, 23]. On the other hand, it is known that the RII
chain and the elliptic Painlevé equation [25] have solutions expressible in terms of the elliptic
hypergeometric function 10E9 [11, 28]. In addition, it was pointed out that the contiguity
relations of the elliptic Painlevé equation are similar to the linear relations of the RII chain [19].
Supported by these evidences, one may believe that a reduction of the RII chain may give rise
to the elliptic Painlevé equation. This work linked the nd-mKdV lattice with the RII chain. We
are now concerned with its relationship to the discrete Painlevé equations. In particular, we
expect that the elliptic Painlevé equation will appear as a reduction of the nd-mKdV lattice.
Acknowledgements
The authors thank Professor Kenji Kajiwara for fruitful comments and the anonymous referees
for their valuable suggestions. This work was supported by JSPS KAKENHI Grant Numbers
11J04105 and 25400110.
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1 Introduction
2 Derivation of the nd-Toda lattice and the nd-LV lattice
2.1 Orthogonal polynomials and the nd-Toda lattice
2.2 Symmetric orthogonal polynomials and the nd-LV lattice
3 Derivation of the RII chain and the nd-mKdV lattice
3.1 RII polynomials and the RII chain
3.2 Symmetric RII polynomials and the nd-mKdV lattice
4 Concluding remarks
References
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