Hypergeometric $tau$-Functions of the $q$-Painleve System of Type $E_7^{(1)}$
We present the $\tau$-functions for the hypergeometric solutions to the $q$-Painlev\'e system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8W_7$. By using the $W(D_5)$ symmetry of the f…
Authors: Tetsu Masuda
Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 5 (2009), 035, 30 pages Hyp e rgeometri c τ -F unction s of the q -P ainlev ´ e System of T yp e E (1) 7 ⋆ T etsu MASUDA Dep artment of P hysics and Mathematics, A oya ma Gakuin University, 5-10-1 F uchinob e, Sagamihar a, Kanagawa, 229-8558, Jap an E-mail: masuda@gem.aoya ma.ac.jp Received Nov em ber 27, 2008, in f inal for m Mar ch 10 , 20 09; Published online March 24 , 20 09 doi:10.38 42/SIGMA.20 09.035 Abstract. W e presen t the τ -functions for the hype r geometric so lutions to the q -Painlev ´ e system of type E (1) 7 in a determinant formula whose e ntries ar e given by the basic hyper - geometric function 8 W 7 . B y using the W ( D 5 ) symmetry of the function 8 W 7 , we construct a se t of tw elve solutions a nd descr ib e the actio n of f W ( D (1) 6 ) on the set. Key wor ds: q -Painlev´ e system; q -hyperg eometric function; W eyl gro up; τ -function 2000 Mathematics Subje ct Classific ation: 33D15; 3 3D05; 3 3D60; 33 E17 1 In tro duction A natural framew ork for discrete P ainlev ´ e equ ations b y mea ns of the geometry of rational surfaces has b een p rop osed by S ak ai [17]. Eac h equation is d ef ined by the group of Cremona transformations on a family of surfaces obtained by blo wing-up at nine p oints on P 2 . Acc ording to the t yp es of rational sur f aces, the discrete Painlev ´ e equations are classif ied in terms of af f ine ro ot systems. A lso, their symmetries are describ ed b y means of af f ine W eyl groups, th e lattice part of whic h gives rise to dif ference equations. F or ins tance, the q -P ainlev ´ e system of typ e E (1) 7 , whic h is the main ob ject of this p ap er, is a discr ete d ynamical sys tem def ined on a family of rational surfaces parameterized by nine-p oint conf igur ations on P 2 suc h th at six p oin ts among them are on a conic and other th r ee are on a line [17]. An explicit expression for the system of q -dif ference equations is giv en by [15] ( f g − tt )( f g − t 2 ) ( f g − 1)( f g − 1) = ( f − b 1 t )( f − b 2 t )( f − b 3 t )( f − b 4 t ) ( f − b 5 )( f − b 6 )( f − b 7 )( f − b 8 ) , ( f g − t 2 )( f g − tt ) ( f g − 1)( f g − 1) = g − t b 1 g − t b 2 g − t b 3 g − t b 4 g − 1 b 5 g − 1 b 6 g − 1 b 7 g − 1 b 8 , (1.1) where t is the indep enden t v ariable and the time evo lution of the dep enden t v ariables is giv en b y g = g ( q t ) and f = f ( t/q ). The parameters b i ( i = 1 , 2 , . . . , 8) satisfy b 1 b 2 b 3 b 4 = q and b 5 b 6 b 7 b 8 = 1. Similarly to the P ainlev ´ e dif feren tial equations, the discrete P ainlev ´ e equations admit par- ticular solutions expressib le in terms of v arious hyp ergeometric functions. Regardin g the q - dif ference P ainlev ´ e equ ations, the h yp ergeometric solutions to those equations ha v e b een con- structed by means of a geometric app roac h and direct linearization of the q -dif ference Riccati ⋆ This paper is a co ntribution t o the Pro ceedings of the W orkshop “Elliptic Integrable S ystems, Isomonodromy Problems, and H yp ergeometric F un ctions” (July 21–25, 2008, MPIM, Bonn, German y). The full collectio n is a v aila ble at http://www.emis .de/journals/SIGMA/Elli ptic-Integrable-Systems.html 2 T. Masuda equations [9, 10]. I n particular, th e Riccati s olution to the system of q -dif ference equations (1.1) is expressed in terms of the q -h yp ergeometric series 8 W 7 ( a 0 ; a 1 , . . . , a 5 ; q , z ) = 8 ϕ 7 a 0 , q a 1 / 2 0 , − q a 1 / 2 0 , a 1 · · · , a 5 a 1 / 2 0 , − a 1 / 2 0 , q a 0 /a 1 , · · · , q a 0 /a 5 ; q , z ! = ∞ X k =0 (1 − a 0 q 2 k ) (1 − a 0 ) ( a 0 ; q ) k ( q ; q ) k 5 Y i =1 ( a i ; q ) k ( q a 0 /a i ; q ) k z k , z = q 2 a 2 0 a 1 a 2 a 3 a 4 a 5 , (1.2) where ( a ; q ) k = k − 1 Q i =0 (1 − aq i ). The purp oses of this p ap er are to prop ose a form ulation for the q -Painlev ´ e system of t yp e E (1) 7 b y means of th e lattice τ -functions and to completely determine th e τ -functions for th e hyper- geometric solutions (h yp ergeometric τ -functions for sh ort) of the system. This pap er is organized as follo w s. In Section 2, we giv e a form ulation for the q -P ainlv ´ e system of t yp e E (1) 7 in terms of the lattice τ - functions. Section 3 is devo ted to a pr eparation for constructing the h yp ergeometric τ -fu nctions. W e d ecomp ose the lattice, eac h of whose elemen ts indicates the τ - function, into a family of six-dimens ional lattices. In Sections 4–6, we constru ct the hyp ergeometric τ -functions. W e f ind that a q -analogue of the doub le gamma function app ears as a normalization factor of the h yp ergeometric τ -functions in Section 4. In Section 5, we f ind that a class of bilinear equations for the lattice τ - functions yields the cont iguit y relations for the q -hypergeometric function 8 W 7 . As is well-kno wn , the q - h yp ergeometric function 8 W 7 p ossesses the W ( D 5 )-symmetry [13]. F rom that, we can construct a set of t w elv e solutions corresp onding to the coset W ( D 6 ) /W ( D 5 ), and describ e the action of f W D (1) 6 on the set of solutions. One of the imp ortan t features of the hyp ergeometric solutions to the con tin uous and discrete P ainlev ´ e equations is that th ey can b e expressed in terms of W ronskians or Casorati deter- minan ts [4, 12, 7, 3, 16 ]. In Section 6, w e sho w that the h yp ergeometric τ -functions of the q -Pa inlev ´ e system of t yp e E (1) 7 are expr essed by “t w o-directional Casorati determinants”. As a consequence, we get an explicit expression for the hyp ergeometric solutions to the q -dif ference P ainlev ´ e equation (1.1), wh ich is prop osed in Corollary 6.1 . 2 The q -Pa inlev ´ e system of t yp e E (1) 7 2.1 The discrete Painlev ´ e system of typ e E (1) 8 A t f irst, we giv e a brief r eview of the form ulation for the discrete P ainlev ´ e system of t yp e E (1) 8 in terms of the lattice τ - functions [8, 11]. Let L = 9 ⊕ i =0 Z e i b e a lattice with a basis { e 0 , e 1 , . . . , e 9 } , and def ine a symmetric bilinear form h , i : L × L → Z by h e 0 , e 0 i = − 1 , h e i , e i i = 1 ( i = 1 , 2 , . . . , 9) , h e i , e j i = 0 ( i, j = 0 , 1 , . . . , 9; i 6 = j ) . Consider the af f ine W eyl group W ( E (1) 8 ) = h s 0 , s 1 , . . . , s 8 i asso ciated with the Dyn kin diagram 1 2 3 4 5 6 7 8 0 ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ The lattice L admits a natural linear action o f W ( E (1) 8 ) def ined by s i . Λ = Λ − h h i , Λ i h i for Λ ∈ L , where h i ( i = 0 , 1 , . . . , 8) are the simple coro ots def ined by h 0 = e 0 − e 1 − e 2 − e 3 and Hyp ergeometric τ -F unctions of the q -P ainlev ´ e System of Typ e E (1) 7 3 h i = e i − e i +1 ( i = 1 , . . . , 8). The canonical cen tral elemen t c = 3e 0 − e 1 − · · · − e 9 is orthogonal to all the simple coro ots h i , and hence W ( E (1) 8 )-in v arian t. The paramete r space for the discrete P ainlev´ e system of type E (1) 8 is the ten-dimens ional v ector sp ace 9 ⊕ i =0 C e i , wh ose co ordinates are denoted b y ε i = h e i , ·i ( i = 0 , 1 , . . . , 9). The ro ot lattice Q ( E (1) 8 ) = 8 ⊕ i =0 Z α i is generated by the simple ro ots α 0 = ε 0 − ε 1 − ε 2 − ε 3 and α i = ε i − ε i +1 ( i = 1 , . . . , 8). The af f ine W eyl group W ( E (1) 8 ) acts on the co ord inate function ε i in a similar w a y to on the basis e i . Th e W ( E (1) 8 )-in v arian t elemen t corresp ondin g to c is giv en by δ = h c, ·i = 3 ε 0 − ε 1 − · · · − ε 9 , which is called th e null ro ot and pla ys th e role of the scaling constan t for dif ference equ ations in the context of the discrete Painlev ´ e equations. F or simplicit y , we denote the ref lecti on s α with resp ect to the ro ot α = ε ij = ε i − ε j or α = ε ij k = ε 0 − ε i − ε j − ε k for i, j, k ∈ { 1 , 2 , . . . , 9 } b y s ij or s ij k , resp ectiv ely . Also, we often use the notation e ij = e i − e j and e ij k = e 0 − e i − e j − e k . F or eac h α ∈ Q E (1) 8 , the action of the translation op erator T α ∈ W E (1) 8 is giv en by [6] T α (Λ) = Λ + h c, Λ i h − 1 2 h h, h i h c, Λ i + h h, Λ i c (Λ ∈ L ) (2.1) b y u sing th e element h ∈ L suc h that α = h h, · i . Note that we h a v e T α T β = T β T α and wT α w − 1 = T w .α for an y w ∈ W E (1) 8 . When α = ε ij or ε ij k , we also denote the translation T α simply by T ij or T ij k , r esp ectiv ely . They can b e expressed by T ij = s il 1 l 2 s il 3 l 4 s l 5 l 6 l 7 s il 3 l 4 s il 1 l 2 s ij , { i, j, l 1 , . . . , l 7 } = { 1 , 2 , . . . , 9 } , T ij k = s l 1 l 2 l 3 s l 4 l 5 l 6 s l 1 l 2 l 3 s ij k , { i, j, k , l 1 , . . . , l 6 } = { 1 , 2 , . . . , 9 } . Let us intro d uce a family of dep endent v ariables τ Λ = τ Λ ( ε ), ε = ( ε 0 , . . . , ε 9 ), indexed by Λ ∈ M , wher e M is the W ( E (1) 8 )-orbit def ined by M = W E (1) 8 . e 1 = { Λ ∈ L | h c, Λ i = − 1 , h Λ , Λ i = 1 } ⊂ L . The action o f W E (1) 8 on the lattice τ -functions τ Λ is d ef ined b y w ( τ Λ ) = τ w . Λ for any w ∈ W E (1) 8 . Th e discrete Pa inlev ´ e system of type E (1) 8 is equiv alent to the ov erdetermined system def ined by the b ilinear equations [ ε j k ][ ε j kl ] τ e i τ e 0 − e i − e l + [ ε k i ][ ε k il ] τ e j τ e 0 − e j − e l + [ ε ij ][ ε ij l ] τ e k τ e 0 − e k − e l = 0 for an y m utually distinct ind ices i, j, k , l ∈ { 1 , 2 , . . . , 9 } , as well as th eir W E (1) 8 -transforms [ w ( ε j k )][ w ( ε j kl )] τ w . e i τ w . (e 0 − e i − e l ) + ( i, j, k )-cyclic = 0 for an y w ∈ W E (1) 8 . Here, [ x ] is a n onzero odd holomorphic function on C satisfying t he Riemann relation [ x + y ][ x − y ][ u + v ][ u − v ] = [ x + u ][ x − u ][ y + v ][ y − v ] − [ x + v ][ x − v ][ y + u ][ y − u ] for any x, y , u, v ∈ C . There are th ree classes of suc h fu n ctions; elliptic, trigonometric and rational. Th ese three cases corresp ond to the three t yp es of d if ference equations, namely , elliptic dif ference, q -dif ference and ordinal dif ference, resp ectiv ely . The lattice part of W E (1) 8 giv es rise to the dif ference P ainlev ´ e equation. 4 T. Masuda 2.2 The q -P ainlev ´ e system of t yp e E (1) 7 Let us prop ose a form ulatio n for the q -Painlev ´ e system of t yp e E (1) 7 b y m eans of the la ttice τ -fu n ctions, using by the n otation in tro duced in the previous su bsection. A d er iv ation of the form ulation is discussed in App endices. The q -P ainlev ´ e system of typ e E (1) 7 is a discrete dynamical system def ined on a f amily of rational surfaces parameterized by nine-p oint conf igur ations on P 2 suc h th at six p oin ts among them are on a conic C and other thr ee are on a line L [17]. Here, we set p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ∈ C and p 7 , p 8 , p 9 ∈ L . In wh at follo ws, the symb ols C and L also m ean the index sets C = { 1 , 2 , 3 , 4 , 5 , 6 } and L = { 7 , 8 , 9 } , resp ectiv ely . And we often use i, j, k , . . . and r , s as the elemen ts of C and L , resp ectiv ely . In this setting, the symmetric groups S C 6 = h s 12 , . . . , s 56 i and S L 3 = h s 78 , s 89 i naturally act on the conf iguration sp ace as the p erm utation of the p oin ts on C and L , resp ectiv ely . Also, the standard Cr emona transform ation with resp ect to ( p 1 , p 2 , p 7 ) is w ell-def ined as a birational act ion on the space. They generate the af f ine W eyl group W ( E (1) 7 ) = h s 12 , s 23 , s 34 , s 45 , s 56 , s 78 , s 89 , s 127 i . The asso ciated Dynkin diagram and its automorphism are realized by ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ e 89 e 78 e 127 e 23 e 34 e 45 e 56 e 12 and π = s 123 s 47 s 58 s 69 , resp ectiv ely . Th us we f ind that the extended af f ine W eyl group f W E (1) 7 = h s 12 , s 23 , s 34 , s 45 , s 56 , s 78 , s 89 , s 127 , π i acts on the conf iguration space. The lattice τ - functions τ Λ = τ Λ ( ε ) for the q -Pai nlev ´ e system of t yp e E (1) 7 are in d exed b y Λ ∈ M E 7 = f W E (1) 7 . e 1 = M C a M L , where M C = { Λ ∈ M | h e 789 , Λ i = 0 } = W E (1) 7 . e 1 , M L = { Λ ∈ M | h e 789 , Λ i = − 1 } = W E (1) 7 . e 7 . The action of f W ( E (1) 7 ) on the lattice τ -functions τ Λ is def ined b y w ( τ Λ ) = τ w . Λ for any w ∈ f W E (1) 7 . The q -Pa inlev ´ e system of t yp e E (1) 7 is equiv alen t to the o v erdetermined system def ined b y the bilinear equations [ ε r s ] τ C e j τ L e 0 − e i − e j = [ ε ij s ] τ L e r τ C e 0 − e i − e r − [ ε ij r ] τ L e s τ C e 0 − e i − e s , [ ε j k ] τ L e r τ C e 0 − e i − e r = [ ε ik r ] τ C e j τ L e 0 − e i − e j − [ ε ij r ] τ C e k τ L e 0 − e i − e k , (2.2) [ ε ij ][ ε ij r ] τ C e k τ C e 0 − e k − e r + ( i, j, k )-cyclic = 0 , [ ε ij ][ ε k l ] τ L e 0 − e i − e j τ L e 0 − e k − e l + ( i, j, k )-cyclic = 0 , (2.3) τ C e i τ C e 0 − e i − e 9 − τ C e j τ C e 0 − e j − e 9 + [ ε ij ][ ε ij 9 ] d L τ L e 7 τ L e 8 = 0 , τ L e 0 − e 1 − e 4 τ L e 0 − e 2 − e 3 − τ L e 0 − e 1 − e 3 τ L e 0 − e 2 − e 4 + [ ε 12 ][ ε 34 ] d C τ C e 5 τ C e 6 = 0 (2.4) for mutually d istinct indices i, j, k , l ∈ C and r , s ∈ L , as w ell as th eir f W ( E (1) 7 )-transforms. The sup erscript C (r esp. L ) denotes that the τ -function is indexed b y Λ ∈ M C (resp. Λ ∈ M L ), and we lea ve it out when it is u nnecessarily . I t is p ossible to f ix the function [ x ] as [ x ] = e ( 1 2 x ) − e ( − 1 2 x ), e ( x ) = e π √ − 1 x , without loss of generalit y . T h e facto rs d L and d C in (2.4) corresp ond to the irred u cible comp onen ts of the an ti-canonical devisor D L = e 0 − e 7 − e 8 − e 9 and D C = 2e 0 − e 1 − · · · − e 6 , resp ectiv ely . These factors are W E (1) 7 -in v arian t and th e action of π is giv en by π : d L ↔ d C . Hyp ergeometric τ -F unctions of the q -P ainlev ´ e System of Typ e E (1) 7 5 The translation op erators with r esp ect to the ro ot vecto rs ε ij , ε r s , ε ij r ∈ Q E (1) 7 are d enoted b y T ij , T r s and T ij r , resp ectiv ely . Also, there exist f ift y six translat ion op er ators that mov e a lattice p oint Λ ∈ M E 7 to its nearest ones. Let us denote suc h op erators by e T ir , e T ij k and e T ir s according to the action on Q E (1) 7 ; e T 17 : ε 78 7→ ε 78 + δ, ε 12 7→ ε 12 − δ, e T 123 : ε 127 7→ ε 127 − δ, ε 34 7→ ε 34 + δ, for instance. W e f ind that these op erators can b e realized as e T ir = T ir s 789 , e T ij k = s 789 T ij k and e T ir s = T ir s s 789 , resp ectiv ely , in terms of the W eyl group W E (1) 8 . Th en, from the f orm ula (2.1), the action on a lattice p oint can b e calculated as e T 17 (e 9 ) = T 17 (e 0 − e 7 − e 8 ) = e 0 − e 1 − e 8 , for example. No te that w e ha v e the relations such as e T 19 e T 178 = 1 and e T 123 e T 456 = 1. The translation op erators with resp ect to the r o ot v ectors can b e expressed by T ij = e T ir e T − 1 j r , T r s = e T − 1 ir e T is and T ij r = e T ij k e T k r . Prop osition 2.1. If the la ttic e τ -functions τ Λ (Λ ∈ M E 7 ) satisfy the biline ar e quations (2.2) and their f W E (1) 7 -tr ansforms, then they also satisfy (2.3) and their f W E (1) 7 -tr ansforms. This is easily verif ied by a direct calculation. F rom this prop osition, w e see th at it is not necessary to consider the b ilinear equations (2.3) for constructing a solution to the q -P ainlev ´ e system of t yp e E (1) 7 . Ho w ev er, as w e will see Section 6, w e use the bilinear equations of t yp e (2.3) in order to get a nicer determin ant formula f or the h yp ergeometric τ -fu nctions. Then , we will treat all t yp es of bilinear equations b elo w, although the discussion b ecomes tec hnically complicated as a consequence. Let us int ro du ce the dep endent v ariables f and g by f = e 1 8 α l − 1 8 α r + 1 4 ε 12 + 1 8 δ e ( 1 4 ε 13 ) τ e 1 τ e 0 − e 1 − e 2 − e ( − 1 4 ε 13 ) τ e 3 τ e 0 − e 2 − e 3 e ( 1 4 ε 13 ) τ e 3 τ e 0 − e 2 − e 3 − e ( − 1 4 ε 13 ) τ e 1 τ e 0 − e 1 − e 2 , g = e 1 8 α r − 1 8 α l + 1 4 ε 12 − 1 8 δ e ( 1 4 ε 23 ) τ e 3 τ e 0 − e 1 − e 3 − e ( − 1 4 ε 23 ) τ e 2 τ e 0 − e 1 − e 2 e ( 1 4 ε 23 ) τ e 2 τ e 0 − e 1 − e 2 − e ( − 1 4 ε 23 ) τ e 3 τ e 0 − e 1 − e 3 with α l = 3 ε 127 + 2 ε 78 + ε 89 and α r = 3 ε 34 + 2 ε 45 + ε 56 . Then , one get the explicit exp r ession for the q -dif ference equations (1.1), a deriv ation of which is d iscussed in Ap p end ix C. 3 A family of six-dimensional lattices and the bilinear equations As a preparation for co nstructing the h yp ergeometric τ - functions, we decomp ose the lat tice M E 7 in to a family of six-dimensional lattices according to the v alue of th e sym metric bilinear form with the coro ot v ector e 89 = e 8 − e 9 ; M E 7 = a n ∈ Z M n , M n = Λ ∈ M E 7 | h Λ , e 89 i = n . P arallel to this decomp osition, let us consider the orthogonal complemen t o f ε 89 in t he ro ot lattice Q E (1) 7 . Then we get the r o ot lattice Q D (1) 6 corresp ondin g to the Dynkin d iagram ❝ ❝ ❝ ❝ ❝ ❝ ❝ ε 12 ε 23 ε 34 ε 45 ε 56 ε 127 δ − ε 567 6 T. Masuda Since w e ha v e ε 127 + ε 12 + 2 ε 23 + 2 ε 34 + 2 ε 45 + ε 56 + ( δ − ε 567 ) = δ , the same δ denotes the n ull r o ot of Q D (1) 6 . The corresp onding simp le ref lections generate the af f in e W eyl group W D (1) 6 = h s 127 , s 12 , s 23 , s 34 , s 45 , s 56 , s δ − ε 567 i . Note that the f inite W eyl group W ( D 6 ) = h s 127 , s 12 , s 23 , s 34 , s 45 , s 56 i includes the s ymmetric group S 6 = h s 12 , s 23 , s 34 , s 45 , s 56 i as a su b- group. In this realization, an automorphism of th e ab o v e Dynkin diagram can b e expressed b y ρ = π s 157 s 168 s 24 s 26 s 35 s 79 whose action on the simple ro ots of Q D (1) 6 is giv en by ρ : ε 12 ↔ δ − ε 567 , ε 127 ↔ ε 56 , ε 23 ↔ ε 45 . The extended af f ine W eyl group f W ( D (1) 6 ) = h s 127 , s 12 , s 23 , s 34 , s 45 , s 56 , s δ − ε 567 , ρ i acts transi- tiv ely on eac h M n . Regarding th e translation op erators, w e hav e e T i 7 , e T ij k ∈ f W D (1) 6 for i, j, k ∈ C = { 1 , 2 , . . . , 6 } , whic h can b e expressed in the form e T α = ρw , w ∈ W D (1) 6 . According to the lo cation of the lattice τ -functions, one can classify the bilinear equa- tions (2.2) in to the follo wing four t yp es: (A) n : Tw o on eac h of M n − 1 , M n and M n +1 , resp ectiv ely . (B) n : F our on M n , and one on M n +1 and M n − 1 , r esp ectiv ely . (C) n : Th ree on M n +1 and M n , resp ectiv ely . (D) n : S ix on M n . The bilinear equ ations of t yp e (C) n are fur ther classif ied int o tw o types. The f irst one is that all of three τ -functions on M n +1 b elong to M C (or M L ), which is denoted by (C) r n . The second is that one of three τ -functions on M n +1 b elongs to M C (or M L ), denoted by (C) i n . Typical bilinear equations are giv en by (A) 0 [ ε 89 ] τ e j τ e 0 − e i − e j = [ ε ij 9 ] τ e 8 τ e 0 − e i − e 8 − [ ε ij 8 ] τ e 9 τ e 0 − e i − e 9 , (B) 0 [ ε 78 ] τ e j τ e 0 − e i − e j = [ ε ij 8 ] τ e 7 τ e 0 − e i − e 7 − [ ε ij 7 ] τ e 8 τ e 0 − e i − e 8 , (C) i 0 [ ε j k ] τ e 8 τ 2e 0 − e i − e j − e k − e 8 − e 9 = [ ε ik 8 ] τ e 0 − e k − e 9 τ e 0 − e i − e j − [ ε ij 8 ] τ e 0 − e j − e 9 τ e 0 − e i − e k , (C) r 0 [ ε ij ] τ e 0 − e i − e j τ e 0 − e k − e 9 + ( i, j, k )-cyclic = 0 , (D) 0 [ ε j k ] τ e 7 τ e 0 − e i − e 7 = [ ε ik 7 ] τ e j τ e 0 − e i − e j − [ ε ij 7 ] τ e k τ e 0 − e i − e k (3.1) for mutually distinct ind ices i, j, k ∈ C . The bilinear equatio ns (2 .3) are also classif ied in a simila r w a y into f our t yp es, eac h of whic h we denote by (A) ′ n , (B) ′ n , (C) ′ n and (D) ′ n to distingu ish them from the bilinear equations (2.2). Typical equations are giv en by (A) ′ 0 [ ε 78 ][ δ − ε 569 ] τ e 9 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 9 + (7 , 8 , 9)-cyclic = 0 , (B) ′ 0 [ ε ij ][ ε k l ] τ e 8 τ 2e 0 − e i − e j − e k − e l − e 8 = [ ε il 8 ][ ε j k 8 ] τ e 0 − e i − e k τ e 0 − e j − e l − [ ε j l 8 ][ ε ik 8 ] τ e 0 − e j − e k τ e 0 − e i − e l , (C) ′ 0 [ ε ij ][ ε ij 9 ] τ e k τ e 0 − e k − e 9 + ( i, j, k )-cyclic = 0 , (D) ′ 0 [ ε ij ][ ε k l ] τ ij τ k l + ( i, j, k )-cyclic = 0 , [ ε ij ][ ε ij 7 ] τ k τ k 7 + ( i, j, k )-cyclic = 0 (3.2) for mutually distinct indices i, j, k , l ∈ C . The bilinear equations (2.4 ) are also classif ied in to the t yp e (A) d n , (B) d n , (C) d n and (D) d n . T ypical equ ations are giv en b y (A) d 0 τ e 8 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 8 − τ e 9 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 9 + [ δ − ε 567 ][ ε 89 ] d C τ e 5 τ e 6 = 0 , (B) d 0 τ e i τ e 0 − e i − e 7 − τ e j τ e 0 − e j − e 7 + [ ε ij ][ ε ij 7 ] d L τ e 8 τ e 9 = 0 , τ e 8 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 8 − τ e 0 − e 1 − e 2 τ e 0 − e 3 − e 4 − [ ε 128 ][ ε 348 ] d C τ e 5 τ e 6 = 0 , (C) d 0 τ e i τ e 0 − e i − e 9 − τ e j τ e 0 − e j − e 9 + [ ε ij ][ ε ij 9 ] d L τ e 7 τ e 8 = 0 , (D) d 0 τ e 0 − e 1 − e 4 τ e 0 − e 2 − e 3 − τ e 0 − e 1 − e 3 τ e 0 − e 2 − e 4 + [ ε 12 ][ ε 34 ] d C τ e 5 τ e 6 = 0 (3.3) for m utually distinct indices i, j ∈ C . Hyp ergeometric τ -F unctions of the q -P ainlev ´ e System of Typ e E (1) 7 7 Lemma 3.1. Any biline ar e quation of typ e (A) 0 c an b e obtaine d by an action of f W D (1) 6 on the first e quation of (3.1) . Also, we have a similar situation r e gar ding e ach c ase of typ e (B) 0 , (C) r 0 , (C) i 0 and (D) 0 , r esp e ctively. Pro of . Any lattice τ - function on M 1 can b e transformed to τ e 8 b y an action of f W D (1) 6 . Searc hing for Λ ∈ M − 1 suc h that h Λ + e 8 , Λ + e 8 i = 0, we f in d that the lattice τ -fu nctions on M − 1 whic h can pair with τ e 8 are τ e 0 − e i − e 8 , τ 2e 0 − e i − e j − e k − e 7 − e 8 and τ c +e i +e 9 − e 7 for mutually distinct indices i, j, k ∈ C . Any of them can b e transformed to τ e 0 − e i − e 8 b y an action of W ( D 6 ). Since τ e 8 is inv arian t und er the action of W ( D 6 ), we f ind th at one of the pairs of the lattice τ -fu n ctions in a bilinear equation of type (A) 0 can b e trans f ormed to τ e 8 τ e 0 − e i − e 8 b y an action of f W D (1) 6 . Not e th at thr ee pairs of the lattice τ -functions in a bilinear equation h av e a common barycen ter. Therefore, th e bilinear equ ations of type (A) 0 including the term τ e 8 τ e 0 − e i − e 8 are reduced to [ ε 89 ] τ e j τ e 0 − e i − e j = [ ε ij 9 ] τ e 8 τ e 0 − e i − e 8 − [ ε ij 8 ] τ e 9 τ e 0 − e i − e 9 , [ ε 89 ] τ e 7 τ e 0 − e i − e 7 = [ ε 79 ] τ e 8 τ e 0 − e i − e 8 − [ ε 78 ] τ e 9 τ e 0 − e i − e 9 , whic h are tr an s formed b y the action of the Dynkin diagram automorphism ρ ∈ f W D (1) 6 to eac h other. The pro of for the other t yp es of bilinear equations is giv en in a similar wa y . F rom this lemma and similar consideration for the bilinear equ ations (3.2) and (3.3), w e immediately get the follo wing pr op osition. Prop osition 3.1. Fix n ∈ Z . 1. Al l the biline ar e qu ations of typ e (A) n c an b e tr ansforme d by the action of f W D (1) 6 to one another. Al so, we ha ve a similar situation r e gar ding e ach c ase of typ e (B) n , (C ) i n , (C) r n and (D) n , r esp e ctively. 2. Al l the biline ar e quations of typ e (A) ′ n c an b e tr an sforme d by the actio n of f W D (1) 6 to one another. Also, we have a similar situation r e gar ding e ach c ase of typ e (B) ′ n and (C) ′ n , r esp e ctively. The set of al l the biline ar e quations of typ e (D) ′ n is de c omp ose d to two orbits by the action of f W D (1) 6 . 3. Al l the biline ar e quations of typ e (A) d n c an b e tr an sforme d by the actio n of f W D (1) 6 to one another. Also, we have a similar situation r e gar ding e ach c ase of typ e (C) d n and (D) d n , r esp e ctively. The set of al l the biline ar e quations of typ e (B) d n is de c omp ose d to two orbits by the action of f W D (1) 6 . Let us discuss the relationships among the ab o v e t yp es of b ilinear equations. Prop osition 3.2. If the lattic e τ -functions satisfy al l the biline a r e quations of typ e (B) n , then they also satisfy those of typ e (A) n ; that is, 1 . (B) n ⇒ (A) n . Similarly, we have 2 . (C) i n ⇒ (C) r n . Mor e over, if τ Λ 6 = 0 for Λ ∈ M n − 1 , we have the f ol lowing: 3 . (C) i n − 1 ⇒ (D) n . 4 . (A) n , (C ) i n − 1 ⇒ (C) i n . 8 T. Masuda Pro of . It is suf f icien t to v erify the statemen t f or a certain n ∈ Z . The f irst and second statemen ts are easily v erif ie d for the case of n = 0. 3. (C) i 0 ⇒ (D) 1 . Let us consider the follo wing bilinear equation [ ε 23 ] τ e 4 τ 2e 0 − e 2 − e 3 − e 4 − e 5 − e 9 = [ ε 349 ] τ e 0 − e 2 − e 9 τ e 0 − e 3 − e 5 − [ ε 249 ] τ e 0 − e 3 − e 9 τ e 0 − e 2 − e 5 of t yp e (C) i 0 . Multiplying this equation b y τ e 0 − e 1 − e 9 and summing up its (1 , 2 , 3)-cyclic p ermu- tations, we get a bilinear equation of t yp e (D) 1 . 4. (A) 0 and (C) i − 1 ⇒ (C) i 0 . Let us consider the follo wing bilinear equation of t yp e (C ) i − 1 [ ε j k ] τ e 9 τ 2e 0 − e i − e j − e k − e 8 − e 9 = [ ε ik 9 ] τ e 0 − e k − e 8 τ e 0 − e i − e j − [ ε ij 9 ] τ e 0 − e j − e 8 τ e 0 − e i − e k . Multiplying b oth right and left-hand s ides b y τ e 8 and using the f irst equ ation of (3.1), w e get τ e 9 × [ ε j k ] τ e 8 τ 2e 0 − e i − e j − e k − e 8 − e 9 = τ e 0 − e i − e j × [ ε ik 8 ] τ e 9 τ e 0 − e k − e 9 + [ ε 89 ] τ e i τ e 0 − e i − e k − { j ↔ k } = τ e 9 × [ ε ik 8 ] τ e 0 − e k − e 9 τ e 0 − e i − e j − [ ε ij 8 ] τ e 0 − e j − e 9 τ e 0 − e i − e k , whic h is equiv al en t to the third equation of (3.1). Also, by not dif f icult b ut tedious p ro cedure, we get the follo wing prop ositions. Prop osition 3.3. If the lattic e τ -functions satisfy al l the biline a r e quations of typ e (B) d n , then they also satisfy those of typ e (A) d n ; that is, 1 . (B) d n ⇒ (A) d n . Similarly, if τ Λ 6 = 0 for Λ ∈ M n − 1 , we have the fol lowing: 2 . (A) n , (C ) d n − 1 ⇒ (C) d n , 3 . (C) i n − 1 , (C) d n − 1 ⇒ (D) d n , 4 . (C) d n − 1 , (C) i n − 1 , (B) n ⇒ (B) d n . Prop osition 3.4. If the lattic e τ -functions satisfy al l the b i line ar e quatio ns of typ e (C) i n , then they also satisfy those of typ e (C) ′ n ; that is, 1 . (C) i n ⇒ (C) ′ n . Similarly, we have 2 . (B) ′ n ⇒ (A) ′ n . Mor e over, if τ Λ 6 = 0 for Λ ∈ M n − 1 , we have the f ol lowing: 3 . (C) ′ n − 1 , (D) n ⇒ (D) ′ n , 4 . (B) ′ n , (C ) i n − 1 ⇒ (B) n . Hyp ergeometric τ -F unctions of the q -P ainlev ´ e System of Typ e E (1) 7 9 4 The construction of the τ -functions on M 0 Hereafter, w e construct the h yp ergeometric τ -functions for the q -P ainlev ´ e system of type E (1) 7 b y imp osing the follo wing b oundary condition τ Λ − 1 = 0 for any Λ − 1 ∈ M − 1 (4.1) and τ Λ 0 6 = 0 for an y Λ 0 ∈ M 0 . In this section, we d iscuss th e constru ction of the τ -fu nctions on the lattice M 0 . First, let us consider the follo wing b ilinear equations of t yp e (A) 0 , (A) ′ 0 and (A) d 0 [ ε 89 ] τ e j τ e 0 − e i − e j = [ ε ij 9 ] τ e 8 τ e 0 − e i − e 8 − [ ε ij 8 ] τ e 9 τ e 0 − e i − e 9 ( i, j ∈ C ) , [ ε 78 ][ δ − ε 569 ] τ e 9 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 9 + (7 , 8 , 9)-cyclic = 0 , [ δ − ε 567 ][ ε 89 ] d C τ e 5 τ e 6 + τ e 8 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 8 − τ e 9 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 9 = 0 . (4.2) The b oun dary condition (4.1) leads u s to [ ε 89 ] = 0 ⇔ ε 89 = ω ∈ Z . (4.3) All the b ilinear equations of t yp e (A) 0 , (A) ′ 0 and (A) d 0 hold und er th e conditions (4.1) and (4.3), since they can b e ob tained b y the action of f W D (1) 6 = h s 127 , s 12 , . . . , s 56 , s δ − ε 567 , ρ i on (4.2) and the co ef f icien t [ ε 89 ] is f W D (1) 6 -in v arian t. Under the b oundary condition (4.1), the b ilinear equations of typ e (B) 0 , (B) ′ 0 and (B) d 0 are expressed in terms of the la ttice τ -fu nctions on M 0 . Typica l equations of these t yp es are giv en b y [ ε 78 ] τ e j τ e 0 − e i − e j = [ ε ij 8 ] τ e 7 τ e 0 − e i − e 7 − [ ε ij 7 ] τ e 8 τ e 0 − e i − e 8 , [ ε ij ][ ε k l ] τ e 8 τ 2e 0 − e i − e j − e k − e l − e 8 = [ ε il 8 ][ ε j k 8 ] τ e 0 − e i − e k τ e 0 − e j − e l − [ ε j l 8 ][ ε ik 8 ] τ e 0 − e j − e k τ e 0 − e i − e l , τ e i τ e 0 − e i − e 7 − τ e j τ e 0 − e j − e 7 + [ ε ij ][ ε ij 7 ] d L τ e 8 τ e 9 = 0 , τ e 8 τ 2e 0 − e 1 − e 2 − e 3 − e 4 − e 8 − τ e 0 − e 1 − e 2 τ e 0 − e 3 − e 4 = [ ε 128 ][ ε 348 ] d C τ e 5 τ e 6 for m utually distinct indices i, j, k , l ∈ C . Th ese are r educed to [ ε 79 ] τ e j τ e 0 − e i − e j = [ ε ij 9 ] τ e 7 τ e 0 − e i − e 7 , τ e i τ e 0 − e i − e 7 = τ e j τ e 0 − e j − e 7 , τ e 0 − e 1 − e 2 τ e 0 − e 3 − e 4 + [ ε 129 ][ ε 349 ] d C τ e 5 τ e 6 = 0 (4.4) and [ ε il 9 ][ ε j k 9 ] τ e 0 − e i − e k τ e 0 − e j − e l = [ ε j l 9 ][ ε ik 9 ] τ e 0 − e j − e k τ e 0 − e i − e l (4.5) due to the conditions (4.1) and (4.3). Ob viously , the equ ation (4.5) can b e deriv ed from the third equation of (4.4) and its S 6 -transforms. Also , it is n ot d if f icult to see that all the bilinear equations of t yp e (D) 0 , (D ) ′ 0 and (D) d 0 can b e d eriv ed fr om the equ ations (4.4) and their f W D (1) 6 - transforms. Then, it is su f f icien t to consider the equations (4.4) and their f W D (1) 6 -transforms for constructing the hyp ergeometric τ -functions on M 0 . Let us consid er a pair of non-zero meromorphic f u nctions ( G ( x ) , F ( x )) satisfying the dif fe- rence equations G ( x + δ ) = ǫ [ x ] G ( x ) and F ( x + δ ) = G ( x ) F ( x ) with a constant ǫ ∈ C ∗ . When Im δ > 0, a t ypical c hoice of suc h functions is giv en by G ( x ) = e − δ 2 x/δ 2 ( u ; q ) ∞ , F ( x ) = e − δ 2 x/δ 3 ( u ; q , q ) ∞ , 10 T. Masuda where u = e ( x ), q = e ( δ ), ( u ; q , q ) = ∞ Q i,j =0 (1 − uq i + j ) and ǫ = − 1. F or other choic es of ( G ( x ) , F ( x ) , ǫ ) , see App endix in [14]. In what follo ws, we f ix a triplet ( G + ( x ) , F + ( x ) , ǫ + ) with a constan t factor ǫ + , n amely w e ha v e G + ( x + δ ) = ǫ + [ x ] G + ( x ) , F + ( x + δ ) = G + ( x ) F + ( x ) . (4.6) Also, w e in tro duce a pair of fun ctions ( G − ( x ) , F − ( x )) by the relations F − ( x ) = F + (2 δ + ω − x ) , G − ( x ) G + ( δ + ω − x ) = 1 . (4.7) Note that these fun ctions satisfy the d if ference equations G − ( x + δ ) = ǫ − [ x ] G − ( x ) , F − ( x + δ ) = G − ( x ) F − ( x ) (4.8) with ǫ − = ( − 1) ω +1 ǫ + . Moreo v er, w e consider a triplet of fun ctions ( A + ( x ) , B + ( x ) , C + ( x )) def ined b y the dif ference equations A + ( x + δ ) A + ( x − δ ) A + ( x ) A + ( x ) = e ( αx + a ) , B + ( x + δ ) B + ( x − δ ) B + ( x ) B + ( x ) = e ( αx + b ) , C + ( x + δ ) C + ( x − δ ) C + ( x ) C + ( x ) = e ( − αx + c ) , (4.9) where a , b , c and α are the complex constan ts satisfying e (2 αδ + 4 b + 2 c ) = ( − 1) ω +1 and ( − 1) ω +1 ǫ 2 + e ( αω + 2 a ) + d C e (5 b + 3 c ) = 0. A t yp ical example of s u c h functions is giv en by A + ( x ) = e ( δ α x/δ +1 3 + a x/δ 2 ). Also, w e introdu ce the functions A − ( x ), B − ( x ) and C − ( x ) b y the relations A − ( x ) = A + (2 δ + ω − x ) , B − ( x ) = B + (2 δ − x ) , C − ( x ) = C + (2 δ − x ) . (4.10) Definition 4.1. F or eac h Λ 0 ∈ M 0 , w e def ine the tw elv e functions τ ( a ; ± ) Λ 0 ( ε ) ( a ∈ C ) by τ ( a ; ± ) Λ 0 ( ε ) = F ± ( ε 79 + ( h e 79 , Λ 0 i + 1) δ ) Y i,j ∈ C ; i 0. W e relate th e v ariables a i to ε j b y a 0 = e ( δ − ε 669 ) , a i = e ( δ − ε i 69 ) ( i = 1 , 2 , . . . , 5) , q = e ( δ ) . (5.1) Since the action of s 457 ∈ W ( D 6 ) = h s 12 , s 23 , s 34 , s 45 , s 56 , s 127 i on the v ariables a i is giv en by s 457 : a 0 7→ q a 2 0 /a 1 a 2 a 3 , a 1 7→ q a 0 /a 2 a 3 , a 2 7→ q a 0 /a 1 a 3 , a 3 7→ q a 0 /a 1 a 2 , a 4 7→ a 4 , a 5 7→ a 5 , w e see that this action leads us to the ab o v e transformation formula for 8 W 7 . Let us in trodu ce the function µ (6) ( ε ) that is i nv arian t under the ac tion of the symmetric group S 5 = h s 12 , s 23 , s 34 , s 45 i ⊂ S 6 and satisf ie s µ (6) ( s 457 ( ε )) µ (6) ( ε ) = g + ( ε 459 ) g + (2 δ − ε 669 ) Q i =4 , 5 g + ( δ + ω − ε i 67 ) g + ( ε 79 ) g + (2 δ − ε 669 − ε 457 ) Q i =4 , 5 g + ( δ + ω − ε i 6 ) , (5.2) where g + ( x ) is give n by G + ( x ) = g + ( x ) ( u ; q ) ∞ with u = e ( x ) and q = e ( δ ). The relation (5.2) means that the function g + (2 δ − ε 669 ) g + ( ε 79 ) Q i ∈ C 6 1 g + ( δ + ω − ε i 6 ) µ (6) ( ε ) is inv arian t un der the action of s 457 . Then, we see that the function µ (6) ( ε ) G + (2 δ − ε 669 ) G + ( ε 79 ) Y i ∈ C 6 G − ( ε i 6 )Φ (6) ( ε ) , (5.3) where Φ (6) ( ε ) = 8 W 7 ( a 0 ; a 1 , a 2 , a 3 , a 4 , a 5 ; q , z ), is inv arian t und er the action of the f inite W eyl group W ( D 5 ) = h s 12 , s 23 , s 34 , s 45 , s 127 i ⊂ W ( D 6 ). 5.2 The con tiguit y r elations for 8 W 7 It is also known that the q -hyp er geometric fu nction Φ (6) = 8 W 7 satisf ies th e follo wing con tiguit y relations [5, 1] ( a 1 − a 2 )(1 − z )Φ (6) = a 1 5 Q i =3 (1 − q a 0 /a 1 a i ) 1 − qa 0 /a 1 Φ (6) | a 1 7→ a 1 /q Hyp ergeometric τ -F unctions of the q -P ainlev ´ e System of Typ e E (1) 7 13 − a 2 5 Q i =3 (1 − q a 0 /a 2 a i ) 1 − q a 0 /a 2 Φ (6) | a 2 7→ a 2 /q , (5.4) ( a 2 − a 1 )(1 − a 0 /a 1 a 2 )Φ (6) = (1 − a 1 )(1 − a 0 /a 1 )Φ (6) | a 1 7→ q a 1 − (1 − a 2 )(1 − a 0 /a 2 )Φ (6) | a 2 7→ q a 2 , (5.5) (1 − a 0 /a 1 )(1 − z )Φ (6) = 5 Q i =1 (1 − a 0 /a i ) (1 − q − 1 a 0 )(1 − a 0 )(1 − q − 1 a 1 ) Φ (6) ( − ) − q − 1 a 1 5 Q i =2 (1 − q a 0 /a 1 a i ) (1 − q − 1 a 1 )(1 − q a 0 /a 1 ) Φ (6) | a 1 7→ a 1 /q , (5.6) Φ (6) | a 1 7→ q a 1 − Φ (6) = q − 1 z (1 − q a 0 )(1 − q 2 a 0 ) 5 Q i =2 (1 − a i ) (1 − a 0 /a 1 ) 5 Q i =1 (1 − q a 0 /a i ) Φ (6) (+) , (5.7) where Φ (6) ( ± ) = Φ (6) | a 0 7→ q ± 2 a 0 ,a 1 7→ q ± 1 a 1 ,...,a 5 7→ q ± 1 a 5 . Noticing that the action of translation op erators e T i 7 ∈ f W D (1) 6 ( i ∈ C ) on the v ariables a i ( i = 0 , 1 , . . . , 5) is giv en by e T i 7 : a i 7→ q − 1 a i , e T 67 : a 0 7→ q − 2 a 0 , a i 7→ q − 1 a i ( i ∈ C 6 ) , w e see that the con tiguit y relations (5.4) and (5.5) can b e rewritten as ( − 1) ω [ ε j k ][ ε 79 ] Φ (6) ( ε ) = Q l ∈ C 6 \{ j,k } [ ε j l 9 ] [ ε j 6 − δ ] Φ (6) e T j 7 ( ε ) − Q l ∈ C 6 \{ j,k } [ ε k l 9 ] [ ε k 6 − δ ] Φ (6) e T k 7 ( ε ) (5.8) and [ ε j k ][ ε j k 9 − δ ] Φ (6) ( ε ) = [ ε j 69 − δ ][ ε j 6 ]Φ (6) e T − 1 j 7 ( ε ) − [ ε k 69 − δ ][ ε k 6 ]Φ (6) e T − 1 k 7 ( ε ) , (5.9) resp ectiv ely , for j, k ∈ C 6 . Similarly , the con tiguit y relations (5.6) and (5.7) are expressed by ( − 1) ω [ ε k 6 ][ ε k 69 ][ ε 79 ]Φ (6) ( ε ) = Q l ∈ C 6 \{ k } [ ε k l 9 ] [ ε k 6 − δ ] Φ (6) e T k 7 ( ε ) − Q l ∈ C 6 [ ε l 6 ] [ δ − ε 669 ][ − ε 669 ] Φ (6) e T 67 ( ε ) (5.10) and Φ (6) ( ε ) = Φ (6) ( e T − 1 k 7 ( ε )) − [3 δ − ε 669 ][2 δ − ε 669 ] Q l ∈ C 6 \{ k } [ ε l 69 − δ ] [ ε k 6 ] Q l ∈ C 6 [ ε l 6 − δ ] Φ (6) e T − 1 67 ( ε ) , (5.11) resp ectiv ely , for k ∈ C 6 . Let us int ro du ce the fu nction Ψ (6) ( ε ) b y Y i,j ∈ C 6 ; i
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