An explicit formula for the discrete power function associated with circle patterns of Schramm type

0.31 MB·English

by  Hisashi Ando

An Explicit Formula for the Discrete Power Function
Associated with Circle Patterns of Schramm Type
Hisashi Ando1, Mike Hay2, Kenji Kajiwara2 and Tetsu Masuda3,
2
1 1Graduate School of Mathematics, Kyushu University,
0
2 744 Motooka, Fukuoka 819-0395, Japan

n 2Institute of Mathematics for Industry, Kyushu University,
a
J 744 Motooka, Fukuoka 819-0395, Japan

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2 3Department of Physics and Mathematics, Aoyama Gakuin University,

] Sagamihara, Kanagawa 229-8558, Japan
I
S
. January 20, 2013
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i
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Abstract
3
v
We present an explicit formula for the discrete power function introduced by Bobenko,
2
1 whichisexpressed intermsofthehypergeometric τfunctions forthesixthPainleve´ equation.
6
Theoriginal definition ofthediscrete powerfunction imposesstrict conditions onthedomain
1
. andthevalueoftheexponent. However,weshowthatonecanextendthevalueoftheexponent
5
to arbitrary complex numbers except even integers and the domain to a discrete analogue of
0
1 the Riemann surface. Moreover, we show that the discrete power function is an immersion
1
whentherealpartoftheexponentisequaltoone.
:
v
i
X
1 Introduction
r
a
Thetheoryofdiscreteanalyticfunctionshasbeendevelopedinrecentyearsbasedonthetheoryof
circle packings or circle patterns, which was initiated by Thurston’s idea of using circle packings
as an approximation of the Riemann mapping[18]. So far many important properties have been
established fordiscreteanalyticfunctions, such as the discrete maximumprincipleand Schwarz’s
lemma[6], the discrete uniformization theorem[15], and so forth. For a comprehensiveintroduc-
tiontothetheoryofdiscreteanalyticfunctions,wereferto[17].
It is known that certain circle patterns with fixed regular combinatorics admit rich structure.
For example, it has been pointed out that the circle patterns with square grid combinatorics intro-
ducedbySchramm[16]andthehexagonalcirclepatterns[5,8,9]arerelatedtointegrablesystems.
Someexplicitexamplesofdiscreteanaloguesofanalyticfunctionshavebeen presentedwhichare
associatedwithSchramm’spatterns: exp(z),erf(z),Airyfunction[16],zγ,log(z)[4]. Also,discrete
analoguesofzγ and log(z)associated withhexagonalcirclepatternsare discussedin[5, 8,9].
1
Among those examples, it is remarkable that the discrete analogue of the power function zγ
associatedwiththecirclepatternsofSchrammtypehasacloserelationshipwiththesixthPainleve´
equation (P )[7]. It is desirable to construct a representation formula for the discrete power
VI
function in terms of the Painleve´ transcendents as was mentioned in [7]. The discrete power
function can be formulated as a solution to a system of difference equations on the square lattice
(n,m) Z2 with a certain initial condition. A correspondence between the dependent variable of
∈
this system and the Painleve´ transcendents can be found in [14], but theformula seems somewhat
indirect. Agafonovhasconstructedaformulafortheradiiofcirclesoftheassociatedcirclepattern
at some special points on Z2 in terms of the Gauss hypergeometric function[3]. In this paper, we
aim to establish an explicit representation formula of the discrete power function itself in terms
of the hypergeometric τ function of P which is valid on Z2 = (n,m) Z2 n,m 0 and for
VI +
{ ∈ | ≥ }
γ C 2Z. Based on this formula, we generalize the domain of the discrete power function to a
∈ \
discreteanalogueoftheRiemann surface.
On the other hand, the fact that the discrete power function is related to P has been used
VI
to establish the immersion property[4] and embeddedness[2] of the discrete powerfunction with
realexponent. Althoughwecannotexpectsuchpropertiesandthusthecorrespondencetoacertain
circle pattern for general complex exponent, we have found a special case of Reγ = 1 where the
discretepowerfunctionis an immersion. Anotherpurposeof thispaperis to provetheimmersion
propertyofthiscase.
This paperis organized as follows. In section 2, we givea brief review of the definition of the
discretepowerfunctionanditsrelationtoP . Theexplicitformulaforthediscretepowerfunction
VI
is given in section 3. We discuss the extension of the domain of the discrete power function in
section 4. In section 5, we show that the discrete power function for Re γ = 1 is an immersion.
Section 6 isdevotedtoconcludingremarks.
2 Discrete power function
2.1 Definition of the discrete power function
For maps, a discrete analogue of conformality has been proposed by Bobenko and Pinkall in the
frameworkofdiscretedifferentialgeometry[10].
Definition 2.1 A map f : Z2 C; (n,m) f is called discrete conformal if the cross-ratio
n,m
→ 7→
with respect toeveryelementaryquadrilateralis equalto 1:
−
(f f )(f f )
n,m − n+1,m n+1,m+1 − n,m+1 = 1. (2.1)
(fn+1,m fn+1,m+1)(fn,m+1 fn,m) −
− −
Thecondition(2.1)isadiscreteanalogueoftheCauchy-Riemannrelation. Actually,asmooth
map f : D C Cisconformal ifandonlyifitsatisfies
⊂ →
(f(x,y) f(x+ǫ,y))(f(x+ǫ,y+ǫ) f(x,y+ǫ))
lim − − = 1 (2.2)
ǫ 0 (f(x+ǫ,y) f(x+ǫ,y+ǫ))(f(x,y+ǫ) f(x,y)) −
→ − −
forall(x,y) D. However,usingDefinition2.1alone, onecannot excludemapswhosebehaviour
∈
is farfrom that ofusual holomorphicmaps. Because of this,an additionalconditionfora discrete
conformalmap has been considered[2, 4, 7, 11].
2
Definition 2.2 A discrete conformal map f is called embedded if inner parts of different ele-
n,m
mentaryquadrilaterals(f , f , f , f )donot intersect.
n,m n+1,m n+1,m+1 n,m+1
An example of an embedded map is presented in Figure 1. This condition seems to require
that f = f is a univalent function in the continuous limit, and is too strict to capture a wide
n,m
class of discrete holomorphic functions. In fact, a relaxed requirement has been considered as
follows[2, 4].
Definition 2.3 A discrete conformal map f is called immersed, or an immersion, if inner parts
n,m
ofadjacentelementaryquadrilaterals(f , f , f , f )aredisjoint.
n,m n+1,m n+1,m+1 n,m+1
See Figure 2 foran exampleofan immersedmap.
Let us givethedefinitionofthediscretepowerfunctionproposedby Bobenko[4, 7, 11].
Definition 2.4 Let f : Z2 C; (n,m) f bea discreteconformalmap. If f isthe solution
+ n,m n,m
→ 7→
tothedifferenceequation
(f f )(f f ) (f f )(f f )
γfn,m = 2n n+1,m − n,m n,m − n−1,m +2m n,m+1 − n,m n,m − n,m−1 (2.3)
f f f f
n+1,m n 1,m n,m+1 n,m 1
− − − −
with theinitialconditions
f = 0, f = 1, f = eγπi/2 (2.4)
0,0 1,0 0,1
for0 < γ < 2, thenwe call f a discretepower function.
∂f
The difference equation (2.3) is a discrete analogue of the differential equation γf = z for
∂z
thepowerfunction f(z) = zγ,whichmeansthattheparameterγcorrespondstotheexponentofthe
discretepowerfunction.
It is easy toget theexplicitformulaofthediscretepowerfunctionform = 0 (or n = 0). When
m = 0, (2.3) is reduced to a three-term recurrence relation. Solving it with the initial condition
f = 0, f = 1, wehave
0,0 1,0
2l l 2k+γ
(n = 2l),
 2l+γ 2k γ
fn,0 =  l 2kY+k=1γ − (n = 2l+1), (2.5)
for n Z . When m = 1 (or n = 1),YkA=1ga2fkon−oγv has shown that the discrete powerfunction can be
+
∈
expressed in terms of the hypergeometric function[3]. One of the aims of this paper is to give an
explicitformulaforthediscretepowerfunction f forarbitrary (n,m) Z2.
n,m +
∈
In Definition2.4,thedomainofthediscretepowerfunctionisrestricted tothe“first quadrant”
Z2, and the exponent γ to the interval 0 < γ < 2. Under this condition, it has been shown that the
+
discrete power function is embedded[2]. For our purpose, we do not have to persist with such a
restriction. Infact, theexplicitformulawewillgiveisapplicabletothecaseγ C 2Z. Regarding
∈ \
thedomain,onecan extendit toadiscreteanalogueoftheRiemann surface.
3
Figure2: Anexampleofthediscrete con-
Figure 1: An example of the embedded
formal map that is not embedded but im-
discreteconformal map.
mersed.
2.2 Relationship to P
VI
In order to construct an explicit formula for the discrete power function f , we will move to
n,m
a more general setting. The cross-ratio condition (2.1) can be regarded as a special case of the
discreteSchwarzian KdV equation
(fn,m − fn+1,m)(fn+1,m+1 − fn,m+1) = pn, (2.6)
(f f )(f f ) q
n+1,m n+1,m+1 n,m+1 n,m m
− −
where p and q are arbitrary functions in the indicated variables. Some of the authors have
n m
constructed various special solutions to the above equation[12]. In particular, they have shown
thatan autonomouscase
(fn,m − fn+1,m)(fn+1,m+1 − fn,m+1) = 1, (2.7)
(f f )(f f ) t
n+1,m n+1,m+1 n,m+1 n,m
− −
where t is independent of n and m, can be regarded as a part of the Ba¨cklund transformations of
P ,and givenspecial solutionsto(2.7)interms oftheτfunctionsofP .
VI VI
We here give a brief account of the derivation of P according to [14]. The derivation is
VI
achievedbyimposingacertainsimilarityconditiononthediscreteSchwarzianKdVequation(2.7)
and the difference equation (2.3) simultaneously. The discrete Schwarzian KdV equation (2.7) is
automaticallysatisfied ifthereexistsafunction v satisfying
n,m
f f = t 1/2v v , f f = v v . (2.8)
n,m n+1,m − n,m n+1,m n,m n,m+1 n,m n,m+1
− −
By eliminatingthevariable f , weget forv thefollowingequation
n,m n,m
t1/2v v +v v = v v +t1/2v v , (2.9)
n,m n,m+1 n,m+1 n+1,m+1 n,m n+1,m n+1,m n+1,m+1
which is equivalent to the lattice modified KdV equation. It can be shown that the difference
equation(2.3)isreduced to
v v v v
n n+1,m − n−1,m +m n,m+1 − n,m−1 = µ ( 1)m+nλ (2.10)
vn+1,m +vn 1,m vn,m+1 +vn,m 1 − −
− −
with γ = 1 +2µ, where λ C is an integration constant. In the following we take λ = µ so that
∈
(2.10)isconsistentwhen n = m = 0 and v +v , 0 , v +v .
1,0 1,0 0,1 0, 1
− −
4
Assume that the dependence of the variable v = v (t) on the deformation parameter t is
n,m n,m
givenby
−2tddt logvn,m = nvvnn++11,,mm +−vvnn−11,,mm +χn+m, (2.11)
−
whereχ = χ (t)isanarbitraryfunctionsatisfyingχ = χ . Thenwehavethefollowing
n+m n+m n+m+2 n+m
Proposition.
v
Proposition2.5 Letq = q = q (t)bethefunctiondefinedbyq = t1/2 n+1,m. Thenqsatisfies
n,m n,m n,m
v
n,m+1
P
VI
d2q 1 1 1 1 dq 2 1 1 1 dq
= + + + +
dt2 2 q q 1 q t! dt! − t t 1 q t! dt
− − − − (2.12)
q(q 1)(q t) t t 1 t(t 1)
+ − − κ2 κ2 +κ2 − +(1 θ2) − ,
2t2(t 1)2 " − 0q2 1(q 1)2 − (q t)2#
∞
− − −
with
1 1
κ2 = (µ ν+m n)2, κ2 = (µ ν m+n)2,
4 − − 0 4 − −
∞
(2.13)
1 1
κ2 = (µ+ν m n 1)2, θ2 = (µ+ν+m+n+1)2,
1 4 − − − 4
where wedenoteν = ( 1)m+nµ.
−
In general, P containsfourcomplexparameters denoted by κ ,κ ,κ and θ. Since n,m Z ,
VI 0 1 +
∞ ∈
a special case of P appears in the above proposition, which corresponds to the case where P
VI VI
admits special solutions expressible in terms of the hypergeometric function. In fact, the special
solutionstoP ofhypergeometrictypearegivenas follows:
VI
Proposition2.6 [13] Definethefunctionτ (a,b,c;t)(c < Z, n Z )by
n ′ +
′ ∈
det(ϕ(a+i 1,b+ j 1,c;t)) (n > 0),
τn′(a,b,c;t) = ( − 1 − 1≤i,j≤n′ (n′′ = 0), (2.14)
with
Γ(a)Γ(b)
ϕ(a,b,c;t) = c F(a,b,c;t)
0 Γ(c)
(2.15)
Γ(a c+1)Γ(b c+1)
+c − − t1 cF(a c+1,b c+1,2 c;t).
1 Γ(2 c) − − − −
−
Here, F(a,b,c;t)istheGausshypergeometricfunction,Γ(x)istheGammafunction,andc andc
0 1
arearbitraryconstants. Then
τ0, 1,0τ 1, 1, 1
q = n′− −n′+−1 − (2.16)
τ 1, 1, 1τ0, 1,0
−n − − n−+1
′ ′
with τk,l,m = τ (a+k+1,b+l+2,c+m+1;t)gives a familyof hypergeometricsolutionsto P
n′ n′ VI
with theparameters
κ = a+n , κ = b c+1+n , κ = c a, θ = b. (2.17)
′ 0 ′ 1
∞ − − −
We callτ (a,b,c;t)orτk,l,m thehypergeometric τfunctionofP .
n′ n′ VI
5
3 Explicit formulae
3.1 Explicit formulae for f and v
n,m n,m
WepresentthesolutiontothesimultaneoussystemofthediscreteSchwarzianKdVequation(2.7)
and thedifference equation(2.3)undertheinitialconditions
f = 0, f = c , f = c tr, (3.1)
0,0 1,0 0 0,1 1
where γ = 2r, andc andc arearbitraryconstants. Weset c = c = 1and t = eπi(= 1)toobtain
0 1 0 1
−
theexplicit formulafortheoriginal discretepowerfunction. Note that τ (b,a,c;t) = τ (a,b,c;t)
n n
′ ′
by the definition. Moreover, we interpret F(k,b,c;t)for k Z as F(k,b,c;t) = 0 and Γ( k) for
>0
∈ −
( 1)k
k Z as Γ( k) = − .
0
∈ ≥ − k!
Theorem 3.1 For(n,m) Z2, thefunction f = f (t)can beexpressed asfollows.
+ n,m n,m
∈
(1) Casewhere n m (or n = n). When n+m is even,we have
′
≤
(r +1) τ ( N, r N +1, r;t)
fn,m = c1tr−nN( r +1N)−1τ ( Nn −+1,−r− N +2,−r+2;t), (3.2)
N n
− − − − −
n+m
where N = and(u) = u(u+1) (u+ j 1)isthePochhammersymbol. Whenn+m
j
2 ··· −
isodd,we have
(r+1) τ ( N +1, r N +1, r;t)
fn,m = c1tr−n( r +1)N−1 τ (nN−+2, r− −N +2, r−+2;t), (3.3)
N 1 n
− − − − − −
n+m+1
where N = .
2
(2) Casewhere n m (or n = m). When n+mis even, wehave
′
≥
(r +1) τ ( N +2, r N +1, r+2;t)
fn,m = c0N( r +1N)−1τm(−N +1,−r−N +2,−r+2;t), (3.4)
N m
− − − − −
n+m
where N = . When n+mis odd,we have
2
(r +1) τ ( N +2, r N +1, r+1;t)
fn,m = c0( r+1)N−1 τm(−N +1,−r−N +2,−r+1;t), (3.5)
N 1 m
− − − − − −
n+m+1
where N = .
2
Proposition3.2 For(n,m) Z2,thefunctionv = v (t)can beexpressed as follows.
+ n,m n,m
∈
(1) Casewhere n m (or n = n). When n+m is even,we have
′
≤
(r) τ ( N +1, r N +1, r+1;t)
vn,m = t−n2( r +N1) τn(−N +1,−r−N +2,−r+2;t), (3.6)
N n
− − − − −
6
6
4
5
3
4
2 3
2
1
1
-3 -2 -1 0 1 -4 -3 -2 -1 0 1
Figure 3: The discrete power function Figure 4: The ordinary power function
withγ = 1+i. z1+i.
n+m
where N = . When n+mis odd,we have
2
τ ( N +1, r N +2, r+1;t)
vn,m = −c1tr−2nτn(−N +2,−r−N +2,−r+2;t), (3.7)
n
− − − −
n+m+1
where N = .
2
(2) Casewhere n m (or n = m). When n+mis even, wehave
′
≥
(r) τ ( N +1, r N +1, r+1;t)
vn,m = t−m2 ( r +N1) τm(−N +1,−r−N +2,−r+2;t), (3.8)
N m
− − − − −
n+m
where N = . When n+mis odd,we have
2
τ ( N +2, r N +2, r+2;t)
vn,m = −c0tm2+1τm(−N +1,−r−N +2,−r+1;t), (3.9)
m
− − − −
n+m+1
where N = .
2
Note that these expressions are applicable to the case where r C Z. A typical example of
∈ \
the discrete power function and its continuous counterpart are illustrated in Figure 3 and Figure
4, respectively. Figure 5 shows an example of the case suggesting multivalency of the map. The
proofoftheabovetheoremand propositionisgiveninthenextsubsection.
Remark3.3 Agafonov has shown that the generalized discrete power function f , under the
n,m
settingofc = c = 1,t = e2iα(0 < α < π)and 0 < r < 1, isembedded[3].
0 1
Remark3.4 Aswementionabove,somespecialsolutionsto(2.7)intermsoftheτfunctionsofP
VI
have been presented[12]. It is easy to show that these solutionsalso satisfya difference equation
which is a deformation of (2.3) in the sense that the coefficients n and m of (2.3) are replaced
7
0.8
0.6
0.4
0.2
-1.0 -0.5 0.5 1.0
-0.2
-0.4
-0.6
-0.8
-1.0
Figure 5: The discrete power function
withγ = 0.25+3.35i.
by arbitrary complex numbers. For instance, a class of solutions presented in Theorem 6 of [12]
satisfies
(α +α +α )f
0 2 4 n,m
(f f )(f f ) (f f )(f f )
= (n α2) n+1,m − n,m n,m − n−1,m (α1 +α2 +α4 m) n,m+1 − n,m n,m − n,m−1 ,
− fn+1,m fn 1,m − − fn,m+1 fn,m 1
− − − − (3.10)
whereα areparametersofP introducedinAppendixA.Settingtheparametersas(α ,α ,α ,α ,α ) =
i VI 0 1 2 3 4
(r,0,0, r+1,0),weseethattheaboveequationisreducedto(2.3)andthatthesolutionsaregiven
−
bythehypergeometricτfunctionsunder theinitialconditions(3.1).
3.2 Proof of the results
In this subsection, we give the proof of Theorem 3.1 and Proposition 3.2. One can easily verify
that f satisfies the initial condition (3.1) by noticing τ (a,b,c;t) = 1. We then show that f
n,m 0 n,m
andv giveninTheorem3.1andProposition3.2satisfytherelation(2.8),thedifferenceequation
n,m
(2.3), thecompatibilitycondition(2.9)and thesimilaritycondition(2.11)bymeans ofthevarious
bilinear relations for the hypergeometric τ function. Note in advance that we use the bilinear
relationsby specializingtheparameters a,band cas
n+m
a = N, b = r N, c = r+1, N = , (3.11)
− − − − 2
when n+m iseven,or
n+m+1
a = r N +1, b = N, c = r +1, N = , (3.12)
− − − − 2
when n+m isodd.
We firstverify therelation (2.8). Notethatwehavethefollowingbilinearrelations
(c 1)τ0, 1, 1τ 1, 1, 1 = (c b 1)tτ0, 1,0τ 1, 1, 2 +bτ0,0,0τ 1, 2, 2,
− n− − −n+1− − − − n+−1 −n − − n −n+1− − (3.13)
(c 1)τ 1, 1, 1τ0, 1, 1 = (c b 1)τ0, 1,0τ 1, 1, 2 +bτ0,0,0τ 1, 2, 2,
−n − − n− − n− −n − − n −n − −
− − −
8
(a b)τ0, 1, 1τ0, 1,0 = aτ 1, 1, 1τ1, 1,0 bτ0,0,0τ0, 2, 1,
m− − m− −m − − m− m m− −
− − (3.14)
(a b)tτ0, 1,0τ0, 1, 1 = aτ 1, 1, 1τ1, 1,0 bτ0,0,0τ0, 2, 1,
− m+−1 m− − −m+1− − m− − m m+−1−
(b a+1)τ0,0,0τ 1, 1, 1 = (b c+1)τ0, 1,0τ 1,0, 1 +(c a)τ0, 1, 1τ 1,0,0,
m −m − − m− m− − m− − −m
− − − (3.15)
(b a+1)τ 1, 1, 1τ0,0,0 = (b c+1)τ0, 1,0τ 1,0, 1 +(c a)τ0, 1, 1τ 1,0,0,
− −m+1− − m − m+−1 m− − − m− − −m+1
for the hypergeometric τ functions, the derivation of which is discussed in Appendix A. Let us
considerthecasewhere n = n. When n+m is even,therelation(2.8)is reduced to
′
rτ[1,1,1]τ[0,1,1] = Ntτ[1,1,2]τ[0,1,0] (r +N)τ[1,2,2]τ[0,0,0],
− n n+1 n+1 n − n n+1 (3.16)
rτ[0,1,1]τ[1,1,1] = Nτ[1,1,2]τ[0,1,0] (r+N)τ[1,2,2]τ[0,0,0],
n n n n n n
− −
wherewedenote
τ[i1,i2,i3] = τ ( N +i , r N +i , r+i ;t), (3.17)
n′ n′ − 1 − − 2 − 3
for simplicity. We see that the relations (3.16) can be obtained from (3.13) with the parameters
specialized as(3.11). In fact, thehypergeometric τfunctionscan berewritten as
τ0, 1, 1 = τ (a+1,b+1,c) = τ ( N +1, r N +1, r+1) = τ[1,1,1], (3.18)
n− − n n − − − − n
forinstance. When n+mis odd,(2.8)yields
rτ[1,2,1]τ[1,1,1] = ( r +N)tτ[1,2,2]τ[1,1,0] Nτ[2,2,2]τ[0,1,0],
− n n+1 − n+1 n − n n+1 (3.19)
rτ[1,1,1]τ[1,2,1] = ( r +N)τ[1,2,2]τ[1,1,0] Nτ[2,2,2]τ[0,1,0],
n n n n n n
− − −
which is also obtained from (3.13) by specializing the parameters as (3.12). Note that the hyper-
geometricτfunctionscan berewrittenas
τ0, 1, 1 = τ (a+1,b+1,c) = τ ( r N +2, N +1, r+1)
n− − n n
− − − − (3.20)
= τ ( N +1, r N +2, r+1) = τ[1,2,1],
n n
− − − −
thistime. Inthecasewheren = m,onecansimilarlyverifytherelation(2.8)byusingthebilinear
′
relations(3.14)and(3.15).
Next,weprovethat (2.3)is satisfied,whichis rewrittenbyusing(2.8)as
f nt 1 m
r n,m = −2 + . (3.21)
− v v 1 +v 1 v 1 +v 1
n,m −n+1,m −n 1,m −n,m+1 −n,m 1
− −
Weusethebilinearrelations
n τ0,0,0τ0, 1, 1 = (b c+1)τ0, 1,0τ0,0, 1 +at 1τ 1, 1, 1τ1,0,0,
′ n′ n′− − − n′−+1 n′−1− − −n′+−1 − n′−1 (3.22)
(a+b c+n +1)τ0,0,0τ0, 1, 1 = aτ 1, 1, 1τ1,0,0 +(b c+1)τ0, 1,0τ0,0, 1,
− ′ n′ n′− − −n′ − − n′ − n′− n′ −
and
τ0,0,0τ 1, 1, 2 = t 1τ 1, 1, 1τ0,0, 1 +τ 1, 1, 1τ0,0, 1,
n −n − − − − −n+1− − n 1− −n − − n −
−
τ0,0,0τ1, 1,0 = τ0, 1,0τ1,0,0 τ0, 1,0τ1,0,0, (3.23)
m m− m− m − m+−1 m 1
−
τ0, 1, 1τ 1,0, 1 = τ 1, 1, 1τ0,0, 1 +τ 1, 1, 1τ0,0, 1,
m− − m− − − −m+1− − m 1− −m − − m −
−
9
fortheproof. TheirderivationisalsoshowninAppendix A.Letusconsiderthecasewheren = n.
′
When n+m iseven,wehave
nτ[1,2,2]τ[1,1,1] = Nτ[1,1,2]τ[1,2,1] +Nt 1τ[0,1,1]τ[2,2,2],
− n n n+1 n 1 − n+1 n 1 (3.24)
mτ[1,2,2]τ[1,1,1] = Nτ[0,1,1]τ[2,2−,2] +Nτ[1,1,2]τ[1,2,1],−
n n n n n n
fromthebilinearrelations(3.22)byspecializingtheparametersa,bandcasgivenin(3.11). These
lead usto
n τ[1,2,2]τ[1,1,1]
v 1 +v 1 = c 1t r+n+1 n n ,
−n+1,m −n−1,m −1 − 2 N τ[n0+,11,1]τ[n1,21,1] (3.25)
−
mτ[1,2,2]τ[1,1,1]
v 1 +v 1 = c 1t r+n n n .
−n,m+1 −n,m−1 − −1 − 2 N τ[n0,1,1]τ[n1,2,1]
By using
τ[1,2,2]τ[0,1,0] = t 1τ[0,1,1]τ[1,2,1] +τ[0,1,1]τ[1,2,1], (3.26)
n n − − n+1 n 1 n n
−
which is obtained from the first relation in (3.23), one can verify (3.21). When n +m is odd, we
havethebilinearrelations
nτ[2,2,2]τ[1,2,1] = ( r +N)τ[1,2,2]τ[2,2,1] +(r+N 1)t 1τ[1,1,1]τ[2,3,2],
− n n − n+1 n 1 − − n+1 n 1 (3.27)
mτ[2,2,2]τ[1,2,1] = (r+N 1)τ[1,1,1]τ−[2,3,2] +( r +N)τ[1,2,2]τ[2,2,1],−
n n n n n n
− −
from (3.22)with(3.12), and
τ[2,2,2]τ[1,1,0] = t 1τ[1,1,1]τ[2,2,1] +τ[1,1,1]τ[2,2,1], (3.28)
n n − − n+1 n 1 n n
−
from the first relation in (3.23). These lead us to (3.21). We next considerthe case where n = m.
′
When n+m iseven,weget thebilinearrelations
mτ[1,2,2]τ[1,1,1] = Nτ[1,1,2]τ[1,2,1] +Nt 1τ[0,1,1]τ[2,2,2],
− m m m+1 m 1 − m+1 m 1 (3.29)
nτ[1,2,2]τ[1,1,1] = Nτ[0,1,1]τ[2,2,2−] +Nτ[1,1,2]τ[1,2,1], −
m m m m m m
and
τ[1,2,2]τ[2,1,2] = τ[1,1,2]τ[2,2,2] τ[1,1,2]τ[2,2,2], (3.30)
m m m m − m+1 m 1
−
from (3.22)and thesecond relation in(3.23), respectively. By usingtheserelations,onecan show
(3.21)inasimilarway tothecasewhere n = n. When n+m isodd,we usethebilinearrelations
′
mτ[2,2,2]τ[1,2,1] = ( r +N)τ[1,2,2]τ[2,2,1] +(r+N 1)t 1τ[1,1,1]τ[2,3,2],
− m m − m+1 m 1 − − m+1 m 1 (3.31)
nτ[2,2,2]τ[1,2,1] = (r+N 1)τ[1,1,1]τ[2−,3,2] +( r+N)τ[1,2,2]τ[2,2,1], −
m m m m m m
− −
and
τ[1,2,1]τ[2,1,1] = τ[1,1,1]τ[2,2,1] +τ[1,1,1]τ[2,2,1], (3.32)
m m − m+1 m 1 m m
−
whichare obtainedfrom (3.22)and thethirdrelationin(3.23), respectively,toshow(3.21).
Wenextgivetheverificationofthecompatibilitycondition(2.9)byusingthebilinearrelations
(c a)τ0, 1, 1τ 1, 1,0 bτ0,0,0τ 1, 2, 1 = (t 1)τ 1, 1, 1τ0, 1,0,
− n′− − n−′+−1 − n′ −n′+−1 − − −n′ − − n′−+1 (3.33)
(c a)tτ0, 1, 1τ 1, 1,0 bτ0,0,0τ 1, 2, 1 = (t 1)τ0, 1,0τ 1, 1, 1.
− n′− − n−′+−1 − n′ −n′+−1 − − n′− −n′+−1 −
10