Table Of ContentA Determinant Formula for a Class of Rational Solutions of Painlev´e
V Equation
1
Tetsu Masuda , Yasuhiro Ohta and Kenji Kajiwara
0
† ‡ †
0 Department of Electrical Engineering, Doshisha University,
2 †
Kyotanabe, Kyoto, 610-0321, Japan
n Department of Applied Mathematics, Faculty of Engineering,
a ‡
J Hiroshima University,
1 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan
3
]
I Abstract
S
. We give an explicit determinant formula for a class of rational solutions of the Painlev´e V equation in
n
termsof theuniversal characters.
i
l
n
[ 1 Introduction and Main Result
1
v It is known that six Painlev´e equations are in general irreducible, namely, their solutions cannot be expressed
6
by “classical functions” in the sense of Umemura [18]. However, it is also known that they admit classical
5
solutions for special values of parameters except for P . Much effort have been made for the investigation of
0 I
1 classical solutions. As a result, it has been recognized that there are two classes of classical solutions. One
0 is transcendental classical solutions expressible in terms of functions of hypergeometric type. Another one is
1 algebraic or rationalsolutions. It is also known that the Painlev´eequations (except for P ) admit action of the
I
0 affine Weyl groups as groups of the B¨acklund transformations. It is remarkable that such classical solutions
/
n are located on special places from a point of view of symmetry in the parameter spaces [12, 13, 14, 15]. For
nli example,PII, PIII andPIV,whosesymmetryis describedbythe affineWeylgroupoftypeA(11), A(11)×A(11) and
A(1), respectively, admit transcendental classicalsolutions on the reflection hyperplanes, and rational solutions
: 2
v on the barycenters of Weyl chambers of the corresponding affine Weyl group.
i
X Umemura et al have investigated the class of solutions on the barycenters of Weyl chambers and found
that (1) these solutions are expressed by some characteristic polynomials generated by the Toda type bilinear
r
a equations, (2) the coefficients of such polynomials admit mysteriouscombinatorialproperties [17, 7, 16]. These
specialpolynomialsaresometimesreferredasYablonskii-Vorob’ev polynomialsforP [19],Okamotopolynomials
II
for P [14], Umemura polynomials for P , P and P .
IV III V VI
One important aspect among such polynomials is that they are expressed as special cases of the Schur
functions. As is well known, the Schur functions are charactersof the irreducible polynomial representationsof
GL(n) and arise as τ-functions of the KP hierarchy [1]. For example, it is known that the special polynomials
forP andP areexpressibleby2-reducedSchurfunctions,andthoseforP by3-reducedSchurfunctions[2,
II III IV
3, 4, 8].
In this paper, we consider P ,
V
d2y 1 1 dy 2 1dy (y 1)2 κ2 y y(y+1)
= + + − κ2 y 0 (θ+1) , (1.1)
dt2 2y y 1 dt − t dt 2t2 ∞ − y − t − 2(y 1)
(cid:18) − (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) −
1
with parameters κ , κ and θ, whose symmetry is described by the affine Weyl group W(A(1)). The aim of
∞ 0 3
this paper is to investigate a class of rational solutions on the barycenters of Weyl chambers and to present an
explicit formula for them.
By the analogy from the known cases, it is naively expected that they are expressed in terms of 4-reduced
Schur functions. However, our formula is expressed by a generalization of Schur functions, which is called the
universal characters and defined as follows [6].
Definition 1.1 Let p =p (t(1)) and q =q (t(2)) , k Z, be two families of polynomials defined by
k k k k
∈
∞ ∞
p ηk =exp t(1)ηj , p =0 for k <0,
k j k
k=0 j=1
X X (1.2)
∞ ∞
q ηk =exp t(2)ηj , q =0 for k <0,
k j k
k=0 j=1
X X
where t(1) = (t(1),t(1), ) and t(2) = (t(2),t(2), ) are the sets of infinite numbers of variables. For any
1 2 ··· 1 2 ···
partitions λ(1) =(λ(1),λ(1), ,λ(1)) and λ(2) =(λ(2),λ(2), ,λ(2)), the universal character S (t(1),t(2))
1 2 ··· n 1 2 ··· m λ(1),λ(2)
is defined as
S (t(1),t(2))=dett q− ,q− , ,q− ,p+ ,p+ , ,p+ , (1.3)
λ(1),λ(2) (cid:18) λ(m2) λ(m2)−1+1 ··· λ(12)+m−1 λ(11)−m λ(21)−m−1 ··· λ(n1)−m−n+1(cid:19)
where
p+ =t(p ,p , ,p ), q− =t(q ,q , ,q ). (1.4)
j j j+1 ··· j+m+n−1 j j j−1 ··· j−m−n+1
Our main result is stated as follows.
Theorem 1.2 Form,n Z ,wedefineafamilyofpolynomialsS =S (t,s)byspecializingS (t(1),t(2))
∈ ≥0 m,n m,n λ(1),λ(2)
as
λ(1) =(n,n 1, ,2,1), λ(2) =(m,m 1, ,2,1), (1.5)
− ··· − ···
t 2s m+n t 2s m+n
(1) (2)
t = + − , t = + − , (1.6)
j −2 j j 2 j
where s is a parameter. For m,n Z , we define S through
<0 m,n
∈
S (t,s)=( 1)m(m+1)/2S (t,s m 1/2),
m,n − −m−1,n − − (1.7)
S (t,s)=( 1)n(n+1)/2S (t,s n 1/2).
m,n m,−n−1
− − −
Then,
S (t,s)S (t,s)
m,n−1 m−1,n
y = , (1.8)
−S (t,s 1)S (t,s+1)
m−1,n m,n−1
−
gives the rational solutions of P (1.1) with the parameters
V
κ =s, κ =s m+n, θ =m+n 1, (1.9)
∞ 0
− −
and
κ = s, κ =s m+n, θ =m+n 1. (1.10)
∞ 0
− − −
Similarly,
2n+1 S (t,s+1/2)S (t,s 1/2)
m,n−1 m,n+1
y = − , (1.11)
2m+1S (t,s 1/2)S (t,s+1/2)
m−1,n m+1,n
−
gives the rational solutions of P (1.1) with the parameters
V
κ =m+1/2, κ =n+1/2, θ =2s m n 1. (1.12)
∞ 0
− − −
2
This result covers all the rational solutions obtained by applying the B¨acklund transformations on the
particular solution of P (1.1),
V
y = 1, κ =s, κ =s, θ = 1. (1.13)
∞ 0
− −
Remark. In ref. [5], Kitaev et al gave a complete classification of rational solutions for P . Our result covers
V
all the rational solutions of the cases (III) and (IV) in their classification. The first half corresponds to the
case (III) and the other does to (IV). Also, Noumi and Yamada presented a determinant formula for a class of
rationalsolutionsofP intermsof2-reducedSchurfunctions [9]. Our resultincludes their formulaasaspecial
V
case, which is explained in Appendix A.
This paper is organized as follows. In Section 2, we give a brief review for the theory of symmetric form
of P [8, 10, 11], which provides us with a clear description of symmetry structure and τ-functions for P . In
V V
Section 3, we construct the rationalsolutions of P by using the theory of symmetric form. Proofof our result
V
is given in Section 4. We mention on the relationship between our result and Yamada’s general determinant
formula[21] of Jacobi-Trudi type in Section 5.
2 Symmetric Form of Painlev´e V Equation
ByusingthetheoryofsymmetricformforP ,itispossibletodescribethestructureofB¨acklundtransformations
V
in a unified manner and to construct particular solutions systematically. In this section, we summarize the
symmetric form of P following refs. [10, 11], and derive bilinear equations satisfied by τ-functions.
V
2.1 Symmetric Form of P
V
P (1.1) is equivalent to the Hamilton system [20]
V
∂H ∂H d
q′ = , p′ = , ′ =t , (2.1)
∂p − ∂q dt
with the Hamiltonian
H =p(p+t)q(q 1)+α qt α pq α p(q 1). (2.2)
2 3 1
− − − −
In fact, putting
κ =α , κ =α , θ =α α 1, (2.3)
∞ 1 0 3 2 0
− −
with
α =1 α α α , (2.4)
0 1 2 3
− − −
we see that equation for y =1 1/q is nothing but P (1.1). Setting
V
−
1 1
f = (t+p), f =√tq, f = p, f =√t(1 q), (2.5)
0 1 2 3
√t −√t −
we obtain the symmetric form of P
V
1
f′ =f f (f f )+ α f +α f ,
0 0 2 1− 3 2 − 2 0 0 2
(cid:18) (cid:19)
1
f′ =f f (f f )+ α f +α f ,
1 1 3 2− 0 2 − 3 1 1 3
(cid:18) (cid:19) (2.6)
1
f′ =f f (f f )+ α f +α f ,
2 2 0 3− 1 2 − 0 2 2 0
(cid:18) (cid:19)
1
f′ =f f (f f )+ α f +α f .
3 3 1 0− 2 2 − 1 3 3 1
(cid:18) (cid:19)
3
We note that the original dependent variable y of P is expressed as
V
f
3
y = . (2.7)
−f
1
In terms of variables f and α , the B¨acklund transformations of P are described by a simple form,
i i V
s (α )= α , s (α )=α +α (j =i 1), s (α )=α (j =i,i 1),
i i i i j j i i j j
− ± 6 ±
α
i
si(fi)=fi, si(fj)=fj (j =i 1), si(fj)=fj (j =i,i 1), (2.8)
± f ± 6 ±
i
π(α )=α , π(f )=f ,
j j+1 j j+1
where the subscripts i = 0,1,2,3 are understood as elements of Z/4Z. These transformations commute with
derivation ′ and satisfy the fundamental relations
s2 =1, s s =s s (j =i,i 1), s s s =s s s (j =i 1),
i i j j i 6 ± i j i j i j ± (2.9)
π4 =1, πs =s π,
j j+1
which means that transformations s (i = 0,1,2,3) generate the affine Weyl group W(A(1)), and s and π
i 3 i
generate its extension including the Dynkin diagram automorphisms.
2.2 τ-Functions and Bilinear Equations
In order to obtain simpler transformation properties, we add a correction term which depends only on t to the
Hamiltonian (2.2). The corrected Hamiltonian h is introduced as
0
α +2α α α +2α +3α
1 2 3 1 2 3
h =f f f f + − f f + f f
0 0 1 2 3 0 1 1 2
4 4
(2.10)
3α +2α +α α 2α α (α +α )2
1 2 3 1 2 3 1 3
f f + − − f f + ,
2 3 3 0
− 4 4 4
and we put h =πj(h ). Then, we have
j 0
α
s (h )=h (i=j), s (h )=h +√t i, π(h )=h . (2.11)
i j j i i i i i+1
6 f
i
We also introduce τ-functions τ (i=0,1,2,3) by
i
τ′
h = i. (2.12)
i
τ
i
Then, defining the action of s (i=0,1,2,3) and π on the τ-functions by
i
τ τ
i−1 i+1
s (τ )=τ (i=j), s (τ )=f , π(τ )=τ , (2.13)
i j j i i i i i+1
6 τ
i
weseethatthe fundamentalrelations(2.9)arepreserved,whichimplies thatthe B¨acklundtransformationscan
be lifted to the level of τ-functions. It should be remarked that we have from (2.7) and (2.13)
τ s (τ )
3 3 3
y = . (2.14)
−τ s (τ )
1 1 1
4
By (2.13), the B¨acklund transformations (2.8) are lead to a set of bilinear equations for τ-functions,
τ s s (τ )=s (τ )s (τ )+α τ τ ,
0 0 1 1 0 0 1 1 0 2 3
τ s s (τ )=s (τ )s (τ ) α τ τ ,
1 1 0 0 0 0 1 1 1 2 3
−
τ s s (τ )=s (τ )s (τ )+α τ τ ,
1 1 2 2 1 1 2 2 1 3 0
τ s s (τ )=s (τ )s (τ ) α τ τ ,
2 2 1 1 1 1 2 2 − 2 3 0 (2.15)
τ s s (τ )=s (τ )s (τ )+α τ τ ,
2 2 3 3 2 2 3 3 2 0 1
τ s s (τ )=s (τ )s (τ ) α τ τ ,
3 3 2 2 2 2 3 3 3 0 1
−
τ s s (τ )=s (τ )s (τ )+α τ τ ,
3 3 0 0 3 3 0 0 3 1 2
τ s s (τ )=s (τ )s (τ ) α τ τ .
0 0 3 3 3 3 0 0 0 1 2
−
Let us define the translation operators T (i=0,1,2,3) by
i
T =πs s s , T =s πs s , T =s s πs , T =s s s π, (2.16)
1 3 2 1 2 1 3 2 3 2 1 3 0 3 2 1
which commute with each other and satisfy T T T T =1. These operators act on parameters α as
1 2 3 0 i
T (α )=α +1, T (α )=α 1, T (α )=α (j =i 1,i), (2.17)
i i−1 i−1 i i i i j j
− 6 −
(1)
and generate the weight lattice of A . In terms of T , τ-functions in (2.15) are expressed as
3 i
τ =T (τ ), τ =T T (τ ), τ =T−1(τ ),
1 1 0 2 1 2 0 3 0 0
s (τ )=T−1T (τ ), s (τ )=T (τ ), s (τ )=T T (τ ),
0 0 0 1 0 1 1 2 0 2 2 1 3 0
s3(τ3)=T3−1(τ0), s0s1(τ1)=T1T2T0−1(τ0), s1s0(τ0)=T2T0−1(τ0), (2.18)
s s (τ )=T T (τ ), s s (τ )=T (τ ), s s (τ )=T−1(τ ),
1 2 2 2 3 0 2 1 1 3 0 2 3 3 2 0
s s (τ )=T T (τ ), s s (τ )=T T−1(τ ), s s (τ )=T T−1T−1(τ ).
3 2 2 1 0 0 3 0 0 1 3 0 0 3 3 1 3 0 0
Furthermore, we can derive bilinear equations of Toda type.
Proposition 2.1 We have
1 1 3α +2α +α
T (τ )T−1(τ )= D2 + 1 2 3t τ τ ,
1 0 1 0 √t 2 T 4 0· 0
(cid:18) (cid:19)
1 1 α 2α α
T (τ )T−1(τ )= D2 1− 2− 3t τ τ ,
2 0 2 0 √t 2 T − 4 0· 0
(cid:18) (cid:19) (2.19)
1 1 α +2α α
T (τ )T−1(τ )= D2 1 2− 3t τ τ ,
3 0 3 0 √t 2 T − 4 0· 0
(cid:18) (cid:19)
1 1 α +2α +3α
T (τ )T−1(τ )= D2 1 2 3t τ τ ,
0 0 0 0 √t 2 T − 4 0· 0
(cid:18) (cid:19)
where D is the Hirota’s differential operator defined by
T
m
d d
Dmf g = f(T)g(T′) , (2.20)
T · dT − dT′
(cid:18) (cid:19) (cid:12)T=T′
(cid:12)
(cid:12)
d d (cid:12)
and =t .
dT dt
Proof. Using (2.8),(2.13) and (2.16), we have
T (τ )T−1(τ )=[f f f +(α +α )f +α f ]τ2. (2.21)
1 0 1 0 1 2 3 1 2 1 1 3 0
5
Noticing that
α +2α α α 2α α
h′ =√t f f f + 1 2− 3f + 1− 2− 1f , (2.22)
0 1 2 3 4 1 4 3
(cid:18) (cid:19)
and f +f =√t, we get the first equation in (2.19). The other equations are obtained in similar way.
1 3
Remark. Bilinearequations(2.15)and(2.19)areoverdeterminedsystemswhentheyareregardedasequations
to determine the τ-functions. However,by construction, the consistency of these equations is guaranteed.
2.3 τ-Cocycles
For simplicity, we introduce a notation,
τ =TkTlTmTn(τ ). (2.23)
k,l,m,n 1 2 3 0 0
For small k,l,m,n, we observe that τ are factorized as
k,l,m,n
k l m n
τ τ τ τ
1 2 3 0
τ =φ τ , (2.24)
k,l,m,n k,l,m,n 0
τ τ τ τ
(cid:18) 0(cid:19) (cid:18) 1(cid:19) (cid:18) 2(cid:19) (cid:18) 3(cid:19)
where φ are some functions of f and α . Conversely, if we define φ by (2.24), it is shown that
k,l,m,n i i k,l,m,n
φ ’s are polynomials in f and α for any k,l,m,n Z [21]. The functions φ are called the τ-
k,l,m,n i i k,l,m,n
∈
cocycles.
It is easy to see from (2.14), (2.18), (2.23) and (2.24) that we have
φ φ
TkTlTmTn(y)= k,l,m,n−1 k,l,m−1,n, for k,l,m,n Z. (2.25)
1 2 3 0 −φ φ ∈
k+1,l,m,n k,l+1,m,n
Moreover,itfollowsfrom(2.13),(2.16),(2.23)and(2.24)thatφ aredeterminedbytherecurrencerelations,
k,l,m,n
φ =T (φ )[f f f (α +α )f α f ]k−l(f f α )l−mfm−n,
k+1,l,m,n 1 k,l,m,n 2 3 0− 3 0 0− 0 2 3 0− 0 0
φ =T (φ )[f f f (α +α )f α f ]l−m(f f α )m−nf1+n−k,
k,l+1,m,n 2 k,l,m,n 3 0 1− 0 1 1− 1 3 0 1− 1 1 (2.26)
φ =T (φ )[f f f (α +α )f α f ]m−n(f f α )1+n−kfk−l,
k,l,m+1,n 3 k,l,m,n 0 1 2− 1 2 2− 2 0 1 2− 2 2
φ =T (φ )[f f f (α +α )f α f ]1+n−k(f f α )k−lfl−m,
k,l,m,n+1 0 k,l,m,n 1 2 3− 2 3 3− 3 1 2 3− 3 3
with φ =1.
0,0,0,0
It is possible to write down the bilinear equations for φ . From (2.18),(2.23) and (2.24), bilinear
k,l,m,n
B¨acklund transformations (2.15) yield to
φ φ =φ φ +(α n+k)φ φ ,
k,l,m,n k+1,l+1,m,n−1 k+1,l,m,n−1 k,l+1,m,n 0 k+1,l+1,m,n k,l,m,n−1
−
φ φ =φ φ (α k+l)φ φ ,
k+1,l,m,n k,l+1,m,n−1 k+1,l,m,n−1 k,l+1,m,n 1 k+1,l+1,m,n k,l,m,n−1
− −
φ φ =φ φ +(α k+l)φ φ ,
k+1,l,m,n k,l+1,m+1,n k,l+1,m,n k+1,l,m+1,n 1 k,l,m,n−1 k,l,m,n
−
φ φ =φ φ (α l+m)φ φ ,
k+1,l+1,m,n k,l,m+1,n k,l+1,m,n k+1,l,m+1,n− 2− k,l,m,n−1 k,l,m,n (2.27)
φ φ =φ φ +(α l+m)φ φ ,
k+1,l+1,m,n k,l−1,m,n k+1,l,m+1,n k,l,m−1,n 2 k,l,m,n k+1,l,m,n
−
φ φ =φ φ (α m+n)φ φ ,
k,l,m,n−1 k+1,l,m,n+1 k+1,l,m+1,n k,l,m−1,n 3 k,l,m,n k+1,l,m,n
− −
φ φ =φ φ +(α m+n)φ φ ,
k,l,m,n−1 k+1,l,m−1,n k,l,m−1,n k+1,l,m,n−1 3 k+1,l,m,n k+1,l+1,m,n
−
φ φ =φ φ (α n+k)φ φ .
k,l,m,n k+1,l,m−1,n−1 k,l,m−1,n k+1,l,m,n−1 0 k+1,l,m,n k+1,l+1,m,n
− −
Similarly, the last two equations in (2.19) are lead to
φ φ
k,l,m+1,n k,l,m−1,n
1 1 α +2α α k l+3m n
= D2 +ω 1 2− 3− − − t φ φ ,
√t 2 T k,l,m,n− 4 k,l,m,n· k,l,m,n
(cid:18) (cid:19) (2.28)
φ φ
k,l,m,n+1 k,l,m,n−1
1 1 α +2α +3α k l m+3n
= D2 +ω 1 2 3− − − t φ φ ,
√t 2 T k,l,m,n− 4 k,l,m,n· k,l,m,n
(cid:18) (cid:19)
6
with
′′ ′′ ′′ ′′
τ τ τ τ
ω =(logτ )′′+k log 1 +l log 2 +m log 3 +n log 0 . (2.29)
k,l,m,n 0
τ τ τ τ
(cid:18) 0(cid:19) (cid:18) 1(cid:19) (cid:18) 2(cid:19) (cid:18) 3(cid:19)
3 Construction of Rational Solutions
In this section, we construct the rational solutions of P by using the results in the previous section.
V
It is obvious that the symmetric form of P (2.6) with (2.4) has a solution,
V
1 1 √t
(α ,α ,α ,α )= s,s, s,s , f = for i=0,1,2,3, (3.1)
0 1 2 3 i
2 − 2 − 2
(cid:18) (cid:19)
which is on the fixed points with respect to the transformationπ2 and is equivalent to the following solution of
P ,
V
y = 1, κ =s, κ =s, θ = 1. (3.2)
∞ 0
− −
ThisistheuniquerationalsolutioninthefundamentalregionoftheaffineWeylgroupW(A(1))intheparameter
3
space,exceptforthespecialcasesoftranscendentalclassicalsolutions[20]. ApplyingB¨acklundtransformations
to the seed solution (3.1), we obtain the family of rational solutions of P . Note that we have
V
1 1
TlTl(α ,α ,α ,α )= s˜,s˜, s˜,s˜ , s˜=s+l, l Z, (3.3)
2 0 0 1 2 3 2 − 2 − ∈
(cid:18) (cid:19)
under the specialization (3.1). Comparing (3.3) with (3.1), we see that the effect of T is absorbed by that
2
of T−1 and shift of the parameter s. Then, we do not need to consider the B¨acklund transformation T for
0 2
constructing the family of rational solutions of P . Taking the initial condition (3.1) into account, we consider
V
the bilinear B¨acklund transformations in terms of the τ-cocycles (2.27). Denoting φ = φ in view of
0,l,m,n l,m,n
the relation T T T T =1, it is easy to derive the following.
1 2 3 0
Lemma 3.1 Under the specialization (3.1), bilinear Ba¨cklund transformations (2.27) are reduced to
φ φ =φ φ +(1/2 s n)φ φ ,
0,m,n 0,m−1,n−2 −1,m−1,n−2 1,m,n 0,m−1,n−1 0,m,n−1
− −
φ φ =φ φ sφ φ ,
−1,m−1,n−1 1,m,n−1 −1,m−1,n−2 1,m,n 0,m−1,n−1 0,m,n−1
−
φ φ =φ φ +sφ φ ,
−1,m−1,n−1 1,m+1,n 1,m,n −1,m,n−1 0,m,n−1 0,m,n
φ φ =φ φ (1/2 s+m)φ φ ,
0,m−1,n−1 0,m+1,n 1,m,n −1,m,n−1− − 0,m,n−1 0,m,n (3.4)
φ φ =φ φ +(1/2 s+m)φ φ ,
0,m−1,n−1 −1,m,n −1,m,n−1 0,m−1,n 0,m,n −1,m−1,n−1
−
φ φ =φ φ (s m+n)φ φ ,
0,m,n−1 −1,m−1,n −1,m,n−1 0,m−1,n 0,m,n −1,m−1,n−1
− −
φ φ =φ φ +(s m+n)φ φ ,
0,m,n−1 −1,m−2,n−1 0,m−1,n −1,m−1,n−2 −1,m−1,n−1 0,m−1,n−1
−
φ φ =φ φ (1/2 s n)φ φ .
0,m,n −1,m−2,n−2 0,m−1,n −1,m−1,n−2 −1,m−1,n−1 0,m−1,n−1
− − −
Moreover, from (2.25), the function
φ φ
0,m,n−1 0,m−1,n
y = , (3.5)
−φ φ
−1,m−1,n−1 1,m,n
solves P (1.1) with parameters
V
κ =s, κ =s m+n, θ =m+n 1. (3.6)
∞ 0
− −
From the recurrence relations (2.26), we observe that φ for small l,m,n are expressed as
l,m,n
(m−n−l−1)(m−n−l)/2
√t
φ = U , (3.7)
l,m,n l,m,n
2
(cid:18) (cid:19)
where U aresome polynomials int ands. Therefore,we nextrewrite (3.4)in terms ofU. The polynomials
l,m,n
U have symmetry described by the following lemma.
l,m,n
7
Lemma 3.2 The polynomials U defined by (3.7) satisfy
l,m,n
U (t,s)=U (t,s+1), U (t,s)=U (t,s 1). (3.8)
1,m,n 0,m,n−1 −1,m,n 0,m,n+1
−
Proof. Lemma 3.2 is proved by considering the Toda type equation for τ-cocycles. Under the specialization
(3.1), the Hamiltonians and τ-functions are calculated as
t2 t2 1 2
h =h = +s2, h =h = + s , (3.9)
0 2 1 3
16 16 2 −
(cid:18) (cid:19)
and
t2 t2
τ =τ =ts2exp , τ =τ =t(1/2−s)2exp , (3.10)
0 2 1 3
32 32
(cid:18) (cid:19) (cid:18) (cid:19)
up to the multiplication by some constants, respectively. Then, Toda type bilinear equations (2.28) yield to
1 1 t2 2s+1 l+3m n
φ φ = D2 + − − − t φ φ ,
l,m+1,n l,m−1,n √t 2 T 8 − 4 l,m,n· l,m,n
(cid:18) (cid:19) (3.11)
1 1 t2 2s+1 l m+3n
φ φ = D2 + − − t φ φ .
l,m,n+1 l,m,n−1 √t 2 T 8 − 4 l,m,n· l,m,n
(cid:18) (cid:19)
Substituting (3.7) into (3.11), we obtain Toda type bilinear equations to be satisfied by U =U (t,s)
m,n 0,m,n
U U =
m+1,n m−1,n
d2U dU 2 dU t 2s+1+3m n
2t m,n U m,n +2 m,nU + − − U2 ,
"(cid:18) dt2 (cid:19) m,n−(cid:18) dt (cid:19) # dt m,n (cid:18)4 − 2 (cid:19) m,n (3.12)
U U =
m,n+1 m,n−1
d2U dU 2 dU t 2s+1 m+3n
2t m,n U m,n +2 m,nU + − U2 ,
"(cid:18) dt2 (cid:19) m,n−(cid:18) dt (cid:19) # dt m,n (cid:18)4 − 2 (cid:19) m,n
with initial conditions
U =U =U =U =1. (3.13)
−1,−1 −1,0 0,−1 0,0
The functions U = U (t,s) are uniquely determined by Toda equations (3.12) from the initial conditions
m,n m,n
(3.13)foranym,n Z. Moreover,weseethatU (t,s)satisfythesameTodaequationsasU (t,s 1),
±1,m,n 0,m,n∓1
∈ ±
respectively, by the similar calculation. Since the initial conditions for U and U are given by
1,m,n −1,m,n
U =U =U =U =1,
1,−1,0 1,0,0 1,−1,1 1,0,1 (3.14)
U =U =U =U =1,
−1,−1,−2 −1,−1,−1 −1,0,−2 −1,0,−1
the lemma is proved.
From Lemma 3.2, bilinear B¨acklund transformations (3.4) are rewritten in terms of U.
Proposition 3.3 Let U =U (t,s) (m,n Z) be polynomials which satisfy the bilinear equations,
m,n m,n
∈
4U U =tU− U+ 2(2s+2n+1)U U ,
m,n+1 m−1,n−1 m−1,n m,n− m−1,n m,n
4U− U+ =tU− U+ 4sU U ,
m−1,n+1 m,n−1 m−1,n m,n− m−1,n m,n
4U− U+ =tU+ U− +4sU U ,
m−1,n m+1,n−1 m,n−1 m,n m,n−1 m,n
4U U =tU+ U− +2(2s 2m 1)U U ,
m−1,n−1 m+1,n m,n−1 m,n − − m,n−1 m,n (3.15)
4U U− =tU− U 2(2s 2m 1)U U− ,
m−1,n−1 m,n+1 m,n m−1,n− − − m,n m−1,n
4U U− =tU− U 4(s m+n)U U− ,
m,n−1 m−1,n+1 m,n m−1,n− − m,n m−1,n
4U U− =tU U− +4(s m+n 1)U− U ,
m+1,n−1 m−1,n m,n m,n−1 − − m,n m,n−1
4U U− =tU U− +2(2s+2n 1)U− U ,
m+1,n m−1,n−1 m,n m,n−1 − m,n m,n−1
8
with
U =U =U =U =1, (3.16)
−1,−1 −1,0 0,−1 0,0
where we denote U± =U (t,s 1). Then,
m,n m,n ±
U (t,s)U (t,s)
m,n−1 m−1,n
y = , (3.17)
−U (t,s 1)U (t,s+1)
m−1,n m,n−1
−
gives the rational solutions of P (1.1) with parameters
V
κ =s, κ =s m+n, θ =m+n 1. (3.18)
∞ 0
− −
4 Proof of Theorem 1.2
In this section, we give the proof for Theorem 1.2.
Definition 4.1 Let p(r) =p(r)(x) and q(r) =q(r)(x) be polynomials defined by
k k k k
∞
xη
p(r)ηk =(1 η)−rexp , p(r) =0 for k <0, (4.1)
k − −1 η k
k=0 (cid:18) − (cid:19)
X
q(r)(x)=p(r)( x), (4.2)
k k −
respectively. For m,n Z , we define a family of polynomials R(r) =R(r) (x) by
≥0 m,n m,n
∈
q(r) q(r) q(r) q(r) q(r) q(r)
1 0 ··· −m+2 −m+1 ··· −m−n+3 −m−n+2
q(r) q(r) q(r) q(r) q(r) q(r)
(cid:12)(cid:12)(cid:12)(cid:12) 3... 2... ·..·.· −m...+4 −m...+3 ·..·.· −m−...n+5 −m−...n+4 (cid:12)(cid:12)(cid:12)(cid:12)
R(r) (x)=(cid:12)(cid:12)(cid:12) q2(rm)−1 q2(rm)−2 ··· qm(r) qm(r−)1 ··· qm(r−)n+1 qm(r−)n (cid:12)(cid:12)(cid:12). (4.3)
m,n (cid:12) p(r) p(r) p(r) p(r) p(r) p(r) (cid:12)
(cid:12) n−m n−m+1 ··· n−1 n ··· 2n−2 2n−1 (cid:12)
(cid:12)(cid:12) ... ... ... ... ... ... ... ... (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12) p(−rn)−m+4 p(−rn)−m+5 ··· p−(rn)+3 p−(rn)+4 ··· p2(r) p3(r) (cid:12)(cid:12)(cid:12)
(cid:12) p(r) p(r) p(r) p(r) p(r) p(r) (cid:12)
(cid:12) −n−m+2 −n−m+3 ··· −n+1 −n+2 ··· 0 1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
For m,n Z , we defi(cid:12)ne R(r) through (cid:12)
<0 m,n
∈
R(r) =( 1)m(m+1)/2R(r) , R(r) =( 1)n(n+1)/2R(r) . (4.4)
m,n − −m−1,n m,n − m,−n−1
Remark. The polynomials p and q (k 0) are essentially the Laguerre polynomials, namely, p(r)(x) =
k k ≥ k
L(r−1)(x). Moreover,R(r) is related to S in Theorem 1.2 as
k m,n m,n
t
R(r) (x)=S (t,s), x= , r =2s m+n. (4.5)
m,n m,n 2 −
9
Proposition 4.2 For m,n Z, R(r) satisfy the following bilinear equations.
m,n
∈
(2n+1)R(r+1) R(r) =xR(r−1) R(r+2) (r+m+n+1)R(r+1) R(r) ,
− m,n+1 m−1,n−1 m−1,n m,n − m−1,n m,n
(2n+1)R(r) R(r+1) =xR(r−1) R(r+2) (r+m n)R(r+1) R(r) ,
− m−1,n+1 m,n−1 m−1,n m,n − − m−1,n m,n
(2m+1)R(r−1) R(r) =xR(r+1) R(r−2)+(r+m n)R(r−1) R(r) ,
m−1,n m+1,n−1 m,n−1 m,n − m,n−1 m,n
(2m+1)R(r) R(r−1) =xR(r+1) R(r−2)+(r m n 1)R(r−1) R(r) ,
m−1,n−1 m+1,n m,n−1 m,n − − − m,n−1 m,n (4.6)
(r) (r−1) (r−2) (r+1) (r) (r−1)
(2n+1)R R =xR R (r m n 1)R R ,
− m−1,n−1 m,n+1 m,n m−1,n− − − − m,n m−1,n
(r−1) (r) (r−2) (r+1) (r) (r−1)
(2n+1)R R =xR R (r m+n)R R ,
− m,n−1 m−1,n+1 m,n m−1,n− − m,n m−1,n
(2m+1)R(r−2) R(r−1) =xR(r) R(r−3) +(r m+n 2)R(r−2)R(r−1) ,
m+1,n−1 m−1,n m,n m,n−1 − − m,n m,n−1
(2m+1)R(r−1) R(r−2) =xR(r) R(r−3) +(r+m+n 1)R(r−2)R(r−1) .
m+1,n m−1,n−1 m,n m,n−1 − m,n m,n−1
Comparing (4.6) with (3.15), we obtain an explicit formula for U =U (t,s) in Proposition3.3.
m,n m,n
Proposition 4.3 We have
U (t,s)=c d S (t,s), m,n Z, (4.7)
m,n m n m,n
∈
where c and d are constants determined by
m n
1
c c = m+ c2 , c =c =1,
m+1 m−1 2 m −1 0
(cid:18) (cid:19) (4.8)
1
d d = n+ d2, d =d =1.
n+1 n−1 − 2 n −1 0
(cid:18) (cid:19)
Proof. Putting
U (t,s)=c d R(r) (x), (4.9)
m,n m n m,n
with
t
x= , r =2s m+n, (4.10)
2 −
we find that the bilinear relations (4.6) become (3.15). Taking (4.5) into account, we obtain Proposition 4.3.
Applying s to the solutions (1.8) with (1.9), we get the solutions (1.8) with (1.10). Then, the first half of
1
Theorem1.2isadirectconsequenceofProposition3.3and4.3. ItiseasytofindthatthelatterhalfofTheorem
1.2 is obtained by applying πs to the solutions (1.8) with (1.9). Therefore, now the proof of Theorem 1.2 is
1
reduced to that of Proposition 4.2.
It is possible to reduce the number of bilinear equations to be provedin (4.6) by the following symmetry of
R(r) (x).
m,n
Lemma 4.4 We have the relations for m,n Z
≥0
∈
R(r) ( x)=R(r) (x), (4.11)
n,m − m,n
R(−r)(x)=( 1)m(m+1)/2+n(n+1)/2R(r) (x). (4.12)
n,m − m,n
Proof. The first relation (4.11) is easily obtained from (4.3). To verify the second relation (4.12), we introduce
polynomials q¯(r) =q¯(r)(x) by
k k
∞
xη
q¯(r)ηk =(1+η)rexp , q¯(r) =0 for k<0. (4.13)
k 1+η k
k=0 (cid:18) (cid:19)
X
10
