Construction of Hypergeometric Solutions to the q-Painlevé Equations

0.14 MB·English

by  Kenji Kajiwara

Construction of Hypergeometric Solutions to the q-Painleve´
Equations
5
0
0 K. Kajiwara1,3, T. Masuda2, M. Noumi2,Y. Ohta2 and Y. Yamada2
2

n 1GraduateSchoolofMathematics,KyushuUniversity,6-10-1Hakozaki,Fukuoka812-8581,Japan
a
2DepartmentofMathematics,KobeUniversity,Rokko,Kobe657-8501,Japan
J
3SchoolofMathematicsandStatisticsF07,UniversityofSydney,Sydney,NSW2006,Australia
1
3
January 30,2005

]
I
S
. Abstract
n
Hypergeometricsolutionstotheq-Painleve´equationsareconstructedbydirectlinearizationofdisrcreteRiccatiequa-
i
l tions.Thedecouplingfactorsareexplicitlydeterminedsothatthelinearsystemsgiverisetoq-hypergeometricequations.
n
[

1 1 Introduction
v
1
Thisarticleisacontinuationofourpreviouswork[1]onthehypergeometricsolutionstotheq-Painleve´equationsinthe
5
followingdegenerationdiagramofaffineWeylgroupsymmetries[2,3]:
0
1 E(1) → E(1) → E(1) → D(1) → A(1) → (A +A )(1) → (A + A1 )(1) (1)
0 8 7 6 5 4 2 1 1 |α|2=14
5
Thelistofq-Painleve´equationswearegoingtoinvestigateisgiveninsection3below. Weremarkthattheseq-Painleve´
0
/ equationswerediscoveredthroughvariousapproachestodiscretePainleve´ equations,includingsingularityconfinement
n analysis, compatibility conditions of linear difference equations, affine Weyl group symmetries and τ-functions on the
i
l lattices. Also, in Sakai’s framework [2], each of these q-Painleve´ equations is constructed in a unified manner as the
n
birationalactionofatranslationofthecorrespondingaffineWeylgrouponacertainfamilyofrationalsurfaces.
:
v
i In[4]wehaveintroducedtheformulationofdiscretePainleve´equationsbasedonthegeometryofplanecurvesonP2.
X
Onthatbasiswewereableinthefirstpartof[1]tofindsuitablecoordinatesforlinearizationoftheq-Painleve´equations
r into three-term relations of hypergeometricfunctions. As a result we obtained the following degeneration diagram of
a
basichypergeometricfunctionscorrespondingto(1):
a
ϕ ;q,z
balanced balanced 1 1 0 ! 0
→ W → → ϕ → ϕ → → ϕ ;q,z
W 8 7 ϕ 2 1 1 1 0 1 1 −q !
10 9 3 2 ϕ ;q,z
1 1 b !
This shows the usefulnessof geometric considerationin the study of particularsolutions of discrete Painlee´ equations.
Inordertodetermineexplicitsolutionstothoseequationswrittenintheliterature,furtherstepsofprecisecomputations
arerequiredsincethevariablestobesolvedarefixedinadvance. Wehaveshownonlytheresultsfortheminthesecond
partof[1]. Thepurposeofthisarticleistopresenttheseexplicitsolutionsincludingsubtlegaugefactorsandtoshowthe
calculationindetailbasedonthedirectlinearizationofdiscreteRiccatiequations.
This article is organized as follows: In section 2, we explain the procedure to construct hypergeometric solutions
throughthe linearizationof discrete Riccati equations. We demonstratethis procedureby taking the case of E(1) as an
7
example. In section 3, we give the list of q-Painleve´ equations and their hypergeometricsolutions, with the data that
arenecessaryforconstructingsolutions. Amongthecasesinthediagram(1)wehaveexcludedthecaseofD(1),sincea
5
completeidentificationofhypergeometricsolutionshasalreadybeengivenin[5].
1
2 Construction of Hypergeometric Solutions
2.1 Discrete RiccatiEquation and Its Linearization
Theq-Painleve´ equationsadmitparticularsolutionscharacterizedbydiscreteRiccatiequationsforspecialvaluesofpa-
rameters.ReductiontodiscreteRiccatiequationhasbeenalreadydoneforalltheq-Painleve´equations[3,6,7,8]. Wealso
notethatsuchspecialsituationshavecleargeometricalmeaning,aswasdiscussedin[1]. Thebasicideaforconstructing
hypergeometricsolutionisasfollows:welinearizethediscreteRiccatiequationstoyieldsecondorderlinearq-difference
equations.Wethenidentifythemwiththethree-termrelationofanappropriatebasichypergeometricseries.
Letusexplainthisprocedureindetail. SupposewehaveadiscreteRiccatiequationoftheform
Az+B
z= , (2)
Cz+D
wherez = z(t)andz = z(qt). Wealsousethenotationz = z(t/q),andsoforth. Moreover,thecoefficientsA, B,C andD
arefunctionsoft. Firstletusputanansatz
F
z= . (3)
G
ThenthediscreteRiccatiequationislinearizedto
F G
= AF+BG, =CF+DG, (4)
H H
whereHisanarbitrarydecouplingfactor.EliminatingGfromeq.(4)wehaveforF thethree-termrelation
H B
F+c F+c F =0, c =− (AB+BD), c = HH(AD−BC). (5)
1 2 1 2
B B
Thethree-termrelationforabasichypergeometricseriesoftentakestheform
V (Φ−Φ)+V Φ+V (Φ−Φ)=0, (6)
1 2 3
where the coefficients V ,V and V are factorized into binomials involving the independent variable and parameters.
1 2 3
Comparingeqs.(5)with(6),wehave
V V
2 =1+c +c , 3 =c . (7)
1 2 2
V V
1 2
WelookforthedecouplingfactorH sothatthesequantitiesfactorize. Wethenidentifythethree-termrelationwiththat
for appropriatehypergeometricfunction. This is done by trial and error with the aid of computeralgebra, but it is not
practically difficult since we already know the hypergeometric function and its three-term relation to appear for each
q-Painleve´equation.
Step1. FindthedecouplingfactorH suchthat
V V
2 =1+c +c , 3 =c , (8)
1 2 2
V V
1 2
factorize. Thenidentifythethree-termrelation
V (F−F)+V F+V (F−F)=0, (9)
1 2 3
withthatforanappropriatehypergeometricfunction.
Similarly,wehaveforGthethree-termrelation
H C
G+d˜ G+d˜ G =0, d˜ =− (DC+CA), d˜ = HH(AD−BC). (10)
1 2 1 2
C C
2
However,usually1+d˜ +d˜ doesnotfactorizeforH obtainedabove.ReplacingGwithκG(κ=kκ),wehave
1 2
1F
z= , (11)
κG
H C HH
G+d G+d G =0, d =− (DC+CA), d = (AD−BC). (12)
1 2 1 2
kC C kk
Wethenlookforksothat1+d +d factorizes,andidentifyeq.(12)withthethree-termrelationofthesamehypergeometric
1 2
functionasF withdifferentparameters.PuttingH/k= K,thisisequivalenttothefollwingprocedure:
Step2. Inthethree-termrelation
K C
G+d G+d G =0, d =− (DC+CA), d = KK(AD−BC), (13)
1 2 1 2
C C
finddecouplingfactorK sothat
U U
2 =1+d +d , 3 =d , (14)
1 2 2
U U
1 2
factorize. Thenidentifythethree-termrelationwiththatforanappropriatehypergeometricfunction.Nowwehave
1 θ Φ H κ
z= 1 , =k, =k, (15)
κ θ Ψ K κ
2
whereΦandΨaresomehypergeometricfunctions,andθ (i=1,2)areconstants(gaugefactors).
i
Finallywedeterminethegaugefactorsθ andθ :
1 2
Step3. Comparethelinearrelations,
θ Φ κθ Ψ
1 =Aθ Φ+κBθ Ψ, 2 =Cθ Φ+κDθ Ψ, (16)
1 2 1 2
H H
withcontiguityrelationsofthehypergeometricfunctionstodetermineθ andθ .
1 2
2.2 AnExample: Caseof E(1)
7
Inthissectionwedemonstratetheconstructionofthehypergeometricsolutiontotheq-Painleve´ equationoftypeE(1) as
7
anexample,followingtheprocedureintheprevioussection.
Beforeproceeding,letusfirstsummarizethedefinitionandterminologyofthebasichypergeometricseries[9]. The
basichypergeometricseries ϕ isgivenby
r s
rϕs ba11,,......,,bars ;q,z!=Xn∞=0(b1;(aq1);nq·)·n·(·b··s;(aqr);n(qq);nq)n h(−1)nq(n2)i1+s−rzn, (17)
(a;q) =(1−a)(1−qa)···(1−qn−1a).
n
Thebasichypergeometricseriesr+1ϕr iscalledbalanced1ifthecondition
qa1a2···ar+1 =b1b2···br, z=q, (18)
issatisfied,andiscalledvery-well-poisedifthecondition
qa1 =a2b1 =···=ar+1br, a2 =qa121, a3 =−qa121, (19)
1The“balanced3ϕ2”inthediagram(1)isduetotheconventionthatwasusedin [10].
3
issatisfied. Avery-well-poisedhypergeometricseriesr+1ϕr isdenotedasr+1Wr:
1 1
r+1Wr(a1;a4,...,ar+1;q,z)=rϕs 1a1,qa112,−qa12,a4...,ar+1 ;q,z. (20)
 a12,−a12,qa1/a4,...,qa1/ar+1 
Nowtheq-Painleve´equationoftypeE(1)isgivenby[2,3]
7
(fg−tt)(fg−t2) (f −b t)(f −b t)(f −b t)(f −b t)
= 1 2 3 4 ,
(fg−1)(fg−1) (f −b )(f −b )(f −b )(f −b )
5 6 7 8
t t t t
g− g− g− g− (21)
(fg−t2)(fg−tt) b ! b ! b ! b !
= 1 2 3 4 ,
(fg−1)(fg−1) 1 1 1 1
g− g− g− g−
b ! b ! b ! b !
5 6 7 8
wheretistheindependentvariableandb (i=1,...,8)areparameterssatisfying
i
t=qt, b b b b =q, b b b b =1. (22)
1 2 3 4 5 6 7 8
Proposition2.1 [3,6]Incaseofb b =b b ,eq.(21)admitsaspecializationtothediscreteRiccatiequation,
1 3 5 7
(tt−1)f +t −(b +b )t+(b +b )
6 8 2 4
g = {−(b +b )+(nb +b )t} f +b b (1−ott), (23)
6 8 2 4 6 8
(t2−1)b b g+t{(b +b )−(b +b )t}
f = 5 7 1 3 5 7 . (24)
{t(b +b )−(b +b )}g+(1−t2)
1 3 5 7
Aswaspointedoutin[1],inthecasesofE(1) thevariables f andgarenotexpressedbyratioofhypergeometricfunctions.
6,7,8
Wechoosethevariableas2
g−t/b
z= 1. (25)
g−1/b
5
ThenthediscreteRiccatiequation(23)and(24)isrewrittenas
Az+B
z= ,
Cz+D
with
A = b b (−b +b t)
1 5 3 5
× b b b q2t3+(b b b −b2b −b b b −b b b −b b b q)qt2
" 4 6 8 1 4 5 4 5 1 4 6 1 4 8 5 6 8
+(b b2+b b b q+b b b q+b b b q−b b b q)t−b b b , (26)
1 4 4 5 6 4 5 8 1 6 8 4 6 8 1 4 5#
B = −(b −b )b2(b b −qb b )t(b −b t)(−1+qt2), (27)
1 4 5 1 4 6 8 5 3
C = −b2b (b −b )(b −b )(b −b t)(−1+qt2), (28)
1 4 5 6 5 8 3 5
D = b b (b −b t)
1 5 5 3
× −b b b qt3+(b b2+b b b q+b b b q+b b b q−b b b q)t2
" 1 4 5 1 4 4 5 6 4 5 8 1 6 8 4 6 8
+(b b b −b2b −b b b −b b b −b b b q)t+b b b . (29)
1 4 5 4 5 1 4 6 1 4 8 5 6 8 4 6 8#
2Thisvariablezshouldbeunderstoodasaratioofτfunctions.
4
Step1. WechoosethedecouplingfactorHas
1
H = . (30)
qb b (b t−b )(b t−b )(b t−b )(b t−b )
1 5 5 3 1 5 4 6 4 8
ThenwehaveforF thethree-termrelation
V (F−F)+V F+V (F−F)=0,
1 2 3
V b (b −b )(b b −qb b )(−1+t)(1+t)(−1+qt2)
2 = 5 3 4 1 4 6 8 ,
V q(b −b t)(−b +b t)(−b +b t)(−b +b t) (31)
1 5 1 6 4 8 4 3 5
V (b −b t)(−b +b t)(−b +b t)(−b +b t)(−1+qt2)
3 = 5 3 1 5 4 6 4 8 .
V (b −b t)(−b +b t)(−b +b t)(−b +b t)(−q+t2)
1 5 1 6 4 8 4 3 5
Thethree-termrelationforthevery-well-poisedbasichypergeometricseries
q2a2
Φ= W a ;a ,a ,a ,a ,a ;q, 0 , (32)
8 7 0 1 2 3 4 5
 a a a a a 
1 2 3 4 5
isgivenby[11]  
U (Φ−Φ)+U Φ+U (Φ−Φ)=0, Φ=Φ| , Φ=Φ| ,
1 2 3 a2→qa2,a3→a3/q a2→a2/q,a3→qa3
a qa qa qa qa
(1−a ) 1− 0 1− 0 1− 0 1− 0 1− 0
2
a ! a ! a a ! a a ! a a !
U1 = 2 2 a 1 3qa 3 4 3 5 , U3 =U1|a2↔a3, (33)
a 1− 2 1− 2
3
a ! a !
3 3
qa2 qa
U = 0 1− 0 (1−a )(1−a )(1−a ).
2 1 4 5
a a a a a a a !
1 2 3 4 5 2 3
Comparingeqs.(31)with(33),weidentifyF with W as
8 7
b b qb b t b b b b
F ∝ W 1 8; 8, 1 , 1 , 2, 4;q, 5 . (34)
8 7
b b b b b t b b b !
3 5 5 5 5 3 3 6
Step2. WechoosethedecouplingfactorKas
1
K = . (35)
b b (qb t−b )(b t−b )(b t−b )(b t−b )
1 5 5 3 1 5 4 6 4 8
ThenwehaveforGthethree-termrelation
X (G−G)+X F+X (G−G)=0,
1 2 3
X b (b −b )(b b −b b )(−1+t)(1+t)(−1+qt2)
2 = 5 3 4 1 4 6 8 ,
X (b −b t)(−b +b t)(−b +b t)(−b +qb t) (36)
1 5 1 6 4 8 4 3 5
X (qb −b t)(−b +b t)(−b +b t)(−b +b t)(−1+qt2)
3 = 5 3 1 5 4 6 4 8 .
X (b −b t)(−b +b t)(−b +b t)(−b +qb t)(−q+t2)
1 5 1 6 4 8 4 3 5
Comparingeqs.(36)with(33),wehave
b b b b t b b b qb
G ∝ W 1 8; 8, 1 , 1 , 2, 4;q, 5 . (37)
8 7
b b b b b t b b b !
3 5 5 5 5 3 3 6
Moreover,fromk =H/K =(1−b /qb t)/(1−b /b t),wehaveκ=1−b /b t. Thereforeweobtain
3 5 3 5 3 5
qa2
W a ;qa ,a ,a ,a ,a ;q, 0
8 7 0 1 2 3 4 5
1  a1a2a3a4a5
z∝ 1− b3 Wa ,a ,a ,a ,a ,a ;q, q2a20 , (38)
b5t 8 7 0 1 2 3 4 5 a1a2a3a4a5
 
5
with
b b b b t b b b q2a2 qb
a = 1 8, a = 8, a = 1 , a = 1 , a = 2, a = 4, 0 = 5. (39)
0 1 2 3 4 5
b b b b b t b b a a a a a b
3 5 5 5 5 3 3 1 2 3 4 5 6
Step3. Letusput
F =θ(qa ,a ,a )Φ(qa ,a ,a ), G =θ(a ,a ,a )Φ(a ,a ,a ), (40)
1 2 3 1 2 3 1 2 3 1 2 3
whereθ(a ,a ,a )isagaugefactortobedetermined. Here,wehaveomittedthedependenceofa ,a anda ,sincethey
1 2 3 0 4 5
arenotrelevanttothecalculation.Thenlinearequations(16)yield
1 θ(qa ,qa ,a /q) A θ(qa ,a ,a )
1 2 3 Φ(qa ,qa ,a /q)= 1 2 3 Φ(qa ,a ,a )+Φ(a ,a ,a ), (41)
κHB θ(a ,a ,a ) 1 2 3 κB θ(a ,a ,a ) 1 2 3 1 2 3
1 2 3 1 2 3
and
κ θ(a ,qa ,a /q) C θ(qa ,a ,a )
1 2 3 Φ(a ,qa ,a /q)= 1 2 3 Φ(qa ,a ,a )+Φ(a ,a ,a ), (42)
κHD θ(a ,a ,a ) 1 2 3 κD θ(a ,a ,a ) 1 2 3 1 2 3
1 2 3 1 2 3
respectively.Now,wehavethecontiguityrelationsforΦ= W [11]
8 7
a q a q a q
a 1− 0 1− 0 1− 0
1
a a ! a a ! a a !
1 3 1 4 1 5 Φ(a /q,a ,a )− a ↔a
a q 1 2 3 1 2
1− 0 (cid:18) (cid:19) (43)
a
1
a2q2
=(a −a ) 1− 0 Φ(a ,a ,a ),
1 2 1 2 3
 a a a a a 
1 2 3 4 5
 
a a a q
(a −1) 1− 0 Φ(a /q,qa ,a )+ 1− 1 1− 0 Φ(a ,a ,a )
2 1 2 3 1 2 3
a ! q ! a !
2 1 (44)
a a q
= a − 1 1− 0 Φ(a /q,a ,a ).
2 1 2 3
q ! a a !
1 2
We denote eqs.(43) and (44) as CR1[a ,a ,a ] and CR2[a ,a ,a ], respectively. Moreover, note that the relations
1 2 3 1 2 3
CR1[a ,a ,a ]andCR2[a ,a ,a ]holdforanypermutationofa ,a anda ,sincetheseparametersareonequalfooting
1 2 3 1 2 3 1 2 3
inΦ= W .
8 7
NowweeliminateΦ(a ,a ,a /q)fromCR1[a ,a ,a ]andCR2[a ,a ,a ]. Shiftinga toqa ,wehavealinearrela-
1 2 3 1 3 2 3 2 1 1 1
tionamongΦ(a ,qa ,a /q),Φ(a ,a ,a )andΦ(a /q,a ,a ),whichshouldcoincidewitheq. (41). Similarly,eliminating
1 2 3 1 2 3 1 2 3
Φ(qa ,a ,a /q) from CR1[qa ,a ,a ] and CR2[a ,a ,a ], we have a linear relation amongΦ(qa ,a ,a ),Φ(a ,a ,a )
1 2 3 1 3 2 3 1 2 1 2 3 1 2 3
and Φ(a ,a ,a /q). Elimination further Φ(a ,a ,a /q) from this relation and CR2[a ,a ,a ] yields a linear relation
1 2 3 1 2 3 3 2 1
amongΦ(a ,qa ,a /q),Φ(qa ,a ,a )andΦ(a ,a ,a ),whichshouldcoincidewitheq.(42). Fromthesecalculations,we
1 2 3 1 2 3 1 2 3
findthatθ(a ,a ,a )shouldsatisfy
1 2 3
a a a b t b
1− 2 1− 3 5 1− 3 1− 4
θ(a ,qa ,a /q) a ! a q ! b ! b qt!
1 2 3 = 0 0 = 8 8 ,
θ(a ,a ,a ) a a a b t b
1 2 3 1− 2 5 1− 3 1− 4 1− 3
a ! a q! b ! b qt!
0 0 8 8 (45)
θ(qa ,a ,a ) a b
1 2 3 =1− 1 =1− 3,
θ(a ,a ,a ) a b
1 2 3 0 1
θ(qa ,qa ,a /q) θ(a ,qa ,a /q) θ(qa ,a ,a )
1 2 3 = 1 2 3 × 1 2 3 ,
θ(a ,a ,a ) θ(a ,a ,a ) θ(a ,a ,a )
1 2 3 1 2 3 1 2 3
whichyield
a a a a b t b b
2 , 3 5, 1 3 , 4 , 3
a q a q a q! qb qb t qb !
θ(a ,a ,a )= 0 0 0 ∞ = 8 8 1 ∞. (46)
1 2 3
a a a b t b
2 5, 3 4 , 3
aq a q! qb qb t!
0 ∞ 8 8 ∞
6
Thereforewearriveatthefinalresult
1 θ(qa ,a ,a ) Φ(qa ,a ,a )
z = 1 2 3 1 2 3
b θ(a ,a ,a ) Φ(a ,a ,a )
1− 3 1 2 3 1 2 3
b t
5
1− b3 8W7 a0;qa1,a2,a3,a4,a5;q, qa20 (47)
= 1− bb31 Wa ,a ,a ,a ,a ,a ;q, a1aq22aa320a4a5.
b5t 8 7 0 1 2 3 4 5 a1a2a3a4a5
 
3 Hypergeometric Solutions
Hypergeometricsolutions to other q-Painleve´ equations can be constructedby the same procedureas that was demon-
stratedin theprevioussection. Insteadofdescribingfullprocedure,we givea list ofequations,solutionsandtheother
datathatarenecessaryforconstructionofsolutions.WenotethatthecaseofD(1)isomittedasmentionedintheintroduc-
5
tion.
3.1 Caseof E(1)
8
3.1.1 EquationandSolution
(1) q-Painleve´equation[3,6,12]
(gst− f)(gst− f)−(s2t2−1)(s2t2−1) P(f,t,m ,...,m )
= 1 7 ,
g g 1 1 P(f,t−1,m ,...,m )
− f − f − 1− 1− 7 1
st !(cid:18)st (cid:19) s2t2! s2t2!
(48)
(fst−g)(fst−g)−(s2t2−1)(s2t2−1) P(g,s,m ,...,m )
= 7 1 ,
f f 1 1 P(g,s−1,m1,...,m7)
−g −g − 1− 1−
st  st ! s2t2! s2t2!
 
where
P(f,t,m ,...,m )= f4−m tf3+(m t2−3−t8)f2
1 7 1 2
(49)
+(m t7−m t3+2m t)f +(t8−m t6+m t4−m t2+1),
7 3 1 6 4 2
andm (k=1,2,...7)aretheelementarysymmetricfunctionsofk-thdegreeinb (i=1,2,...,8)with
k i
b b ···b =1. (50)
1 2 8
Moreover,
t=qt, t=q12s. (51)
(2) Constraintonparameters[6]
qb b b b =1, b b b b =q. (52)
1 3 5 7 2 4 6 8
(3) Hypergeometricsolution
s b
g− + 1
b s ! Φ(q4a ;a ,q2a ,...,q2a )
z= 1 =λ 0 1 2 7 , (53)
s b Φ(a ;a ...,a )
g− + 8 0 1 7
b s !
8
7
whereΦisdefinedintermsofthebalanced W seriesby
10 9
Φ(a ;a ,...,a )= W (a ;a ,...,a ;q2,q2)
0 1 7 10 9 0 1 7
a q2a
q2a , 7;q2 a , 7;q2
+ 0 a0 !∞ 6 k ak !∞ W a27;a1a7,...,a6a7,a ;q2,q2 . (54)
aa70,qa2a027;q2∞Yk=1 qa2ak0,aaka07;q2!∞ 10 9a0 a0 a0 7 
Here,a (i=0,1,...,7)andλaregivenby
i
1 q2 s2
a = , a = , a = ,
0 qb b b2 1 b b t2 2 b b
1 2 8 2 8 2 8 (55)
b b
a = i (i=3,5,7), a = i (i=4,6),
i i
b b
8 1
and
b b b b b
1− 4 6 1−q2 4 6 (1−b b t2)(1−b b t2)(1−b b t2) 1− i
3 5 3 7 5 7
b b ! b b ! b !
λ= b1b4b6 1 8 1 8 i=Y2,4,6 1 , (56)
b s2 s2 q2s2 b b q b b
8 1− 1− 1− 4 1− 6 1− 1− 4 6
b b ! b b ! b ! b ! b b s2! b b !
1 8 1 8 8 8 1 8 i=Y3,5,7 1 i
respectively.
3.1.2 Data
(1) Riccatiequation
1 f g fg 1 f g fg
(cid:12)(cid:12) 1 f1 g1 f1g1 (cid:12)(cid:12)=0, (cid:12)(cid:12) 1 f8 g8 f8g8 (cid:12)(cid:12)=0, (57)
(cid:12)(cid:12)(cid:12) 1 f3 g3 f3g3 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) 1 f6 g6 f6g6 (cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) 1 f5 g5 f5g5 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) 1 f4 g4 f4g4 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
where (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
1 s b
f =bt+ , g = + i. (58)
i i i
bt b s
i i
g−g
TheRiccatiequationforz= 1 isgivenby
g−g
8
Az+B
z= , (59)
Cz+D
B=−f g g d′ ,
35 13 15 1468
C = f g g d ,
46 48 68 1358 (60)
D=−f f f g g g g ,
35 46 18 13 15 48 68
∆= AD−BC = f f f f f f g g g g g g g g ,
13 35 15 46 48 68 13 15 35 18 46 48 68 18
where f = f − f and
ij i j
1 f g f g 1 f g f g
1 1 1 1 1 1 1 1
d =(cid:12)(cid:12) 1 f3 g3 f3g3 (cid:12)(cid:12), d′ =(cid:12)(cid:12) 1 f4 g4 f4g4 (cid:12)(cid:12), (61)
1358 (cid:12)(cid:12)(cid:12) 1 f5 g5 f5g5 (cid:12)(cid:12)(cid:12) 1468 (cid:12)(cid:12)(cid:12) 1 f6 g6 f6g6 (cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) 1 f8 g8 f8g8 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) 1 f8 g8 f8g8 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
respectively. (cid:12) (cid:12) (cid:12) (cid:12)
8
(2) Three-termandcontiguityrelationsforhypergeometricfunction[13]
(a) Three-termrelation
U (Φ−Φ)+U Φ+U (Φ−Φ)=0, (62)
1 2 3
where
a q2a
a (1−a ) 1− 0 1− 0
1 2
a ! a !
U = 2 2 ,
1
q2a 7 q2a
(1− 2) 1− 0
a a a ! (63)
1 Yj=3 1 j
q2a 7
U =−(a −a ) 1− 0 (1−a ), U =U | ,
2 1 2 a a ! j 3 1a1↔a2
1 2 Yj=3
Φ=Φ(a ;a /q2,q2a ,a ,...,a ), Φ=Φ(a ;q2a ,a /q2,a ,...,a ), (64)
0 1 2 3 7 0 1 2 3 7
andΦisdefinedbyeq.(54).
(b) Contiguityrelations
Φ(a ;a /q2,q2a ,a ,...,a )−Φ(a ;a ,a ,a ,...,a )
0 1 2 3 7 0 1 2 3 7
=V Φ(q4a2;a ,q2a ,...,q2a ), (65)
1 0 1 2 7
V Φ(q4a2;a ,q2a ,q2a ,...,q2a )−V Φ(q4a2;q2a ,a ,q2a ,...,q2a )
2 0 1 2 3 7 3 0 1 2 3 7
=V Φ(a ;a ,a ,a ,...,a ), (66)
4 0 1 2 3 7
where
q2a q2a a a 7
0 1− 2 1− 1 2 (1−q2a )(1−q4a ) (1−a )
a a ! q2a ! 0 0 j
V = 2 1 0 Yj=3 ,
1
q2a q4a a q2a 7 q2a
1− 0 1− 0 1− 0 1− 0 1− 0
a ! a ! a ! a ! a !
1 1 2 2 Yj=3 j
7 q2a
a2(1−a ) 1− 0
1 2 a a ! (67)
V = Yj=3 1 j , V =V | ,
2 q2a q4a 3 2a1↔a2
1− 0 1− 0
a ! a !
1 1
a 7 q2a
a 1− 2 1− 0
1
a ! a !
V = 1 Yj=3 j .
4 (1−q2a )(1−q4a )
0 0
(3) Decouplingfactors
D f 1 1
H = =− 18 , K = =− , (68)
∆ f f f f g g g g D f f f g g g g
13 15 48 68 35 18 46 18 35 46 18 13 15 48 68
sothat
k= H = D2 = f35f46f128g13g15g48g68. (69)
K ∆ f f f f g g g
13 15 48 68 35 46 18
(4) Identification
1 F
z= , F ∝Φ(q4a ;a ,q2a ,...,q2a ), G ∝Φ(a ;a ...,a ), κ=kκ, (70)
κ G 0 1 2 7 0 1 7
wherea (i=0,...,7)aregivenbyeq.(55).
i
9
(5) Gaugefactors
Putting
F =θ(q4a ;a ,q2a ,...,q2a )Φ(q4a ;a ,q2a ,...,q2a ),
0 1 2 7 0 1 2 7
(71)
G =θ(a ;a ...,a )Φ(a ;a ...,a ),
0 1 7 0 1 7
wehave:
θ(a ;a /q2,a q2,...,a ) 1θ(q4a ;a ,q2a ,...,q2a )
0 1 2 7 =1, 0 1 2 7 =λ. (72)
θ(a ;a ...,a ) κ θ(a ;a ...,a )
0 1 7 0 1 7
3.2 Caseof E(1)
6
3.2.1 EquationandSolution
(1) q-Painleve´equation[3,14,15]
(f −b t)(f −b t)(f −b t)(f −b t)
(fg−1)(fg−1)=tt 1 2 3 4 ,
t
(f −b t) f −
5 b !
5
1 1 1 1
g− g− g− g− (73)
b ! b ! b ! b !
(fg−1)(fg−1)=t2 1 2 3 4 ,
t
(g−b t) g−
6
b !
6
t=qt, b b b b =1.
1 2 3 4
(2) Constraintonparameters[3,15]
b b =b b . (74)
1 2 5 6
(3) Hypergeometricsolution
1 b
g− 1− 3
b b Φ(qa,b,c,d,e)
z= 1 = 1 , (75)
g−tb b b b t Φ(a,b,qc,d,e)
6 1− 1 2 3
b
5
whereΦisthebalanced ϕ seriesdefinedby
3 2
a,b,c de
Φ(a,b,c,d,e)= ϕ ;q, , (76)
3 2 d,e abc!
with
a= b3b5, b= b3, c=b2b b , d =qb3b25, e=qb b b2. (77)
t b 1 2 3 b 1 2 3
2 2
3.2.2 Data
(1) Riccatiequation[3,15]
b t
1+ 5 (f −b −b )
b b 1 3 1+b t(b b g−b −b )
g= 1 2 , f = 6 3 4 3 4 . (78)
f −tb5 g−tb6
TheRiccatiequationfor
g−1/b
z= 1, (79)
g−tb
6
10