Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

Rational solutio ns of the d iscrete time T o d a l att ice and the altern ate di s crete P ainlev ´ e I I equation Alan K. Common and Andrew N. W. Hone † † Institute of Mathema tics, Statistics & Actuarial Science, Universit y of Kent, Canterbury CT2 7NF, UK E-mail: A.N.W.H one@k ent.ac.uk Abstract. The Y ablonskii-V o rob’ev po lynomials y n ( t ), which are defined b y a sec o nd order bilinear differential-difference equation, provide rational solutions of the T o da lattice. They are als o po lynomial tau-functions for the ra tional solutions of the second Painlev ´ e equation ( P I I ). Here we define t wo-v ar iable p olynomia ls Y n ( t, h ) on a lattice with spacing h , by considering ra tional solutions of the discre te time T o da lattice as int ro duced by Suris. These p olynomials are shown to have many pr o p erties that are analogo us to those of the Y ablo nskii-V oro b’ev po lynomials, to which they reduce when h = 0. They a lso provide rationa l solutions for a particular discr etisation of P I I , namely the so called alternate discr ete P I I , and this connection leads to a n expres s ion in terms of the Umemura po lynomials for the third Painlev´ e equation ( P I I I ). It is shown that B¨ acklund transfor mation for the alternate discrete Painlev ´ e equation is a symplectic map, and the shift in time is also symplectic. Finally w e present a Lax pair for the alterna te discrete P I I , which r ecov ers J im bo and Miw a’s Lax pair for P I I in the contin uum limit h → 0. Submitted to: J. Phys. A: Math. The or. R ational solutions of discr ete T o da and al t - dP I I 2 1. In tro duction The T o da lattice d 2 x n dt 2 = e x n − 1 − x n − e x n − x n +1 , n ∈ Z (1) w as the first in tegrable differen tia l-difference equation to be disco v ered [1]. The Y ablonskii-V orob’ev p olynomials [2 , 3] yield rational solutio ns of b oth the T o da lattice and the second Painlev ´ e t r a nscenden t ( P I I ), since the tau-f unctions of P I I satisfy the bilinear form of the T o da lattice. In a previous work [4] o ne of the a ut ho rs obtained an expression for solutions of the T o da lat t ice as ratios of Hank el determinants, b y using the asso ciated Lax pair to construct con tin ued fraction solutions to a sequence of Riccati equations. This in turn led to an express ion for the Y ablonskii-V orob’ev p oy nomials as Hank el determinan t s [5], equiv alen t to that disco v ered more recen tly [6] (see also [7]). Here w e will consider the case when the time ev olution b ecomes discrete. Our approac h is t o start from the Lax pair for the discrete T o da lat tice (dTL) g iv en by Suris [8]. In section 2 w e presen t this construction and deriv e p o lynomials Y n ( t, h ) in t wo v a r iables t, h , whic h satisfy relations at a discrete set of p o in ts t = t 0 + mh , m ∈ Z (where t 0 is arbitra ry). These p olynomials tend to the Y ablonskii-V orob’ev p olynomials y n ( t ) as the spacing h → 0, so that Y n ( t, 0) = y n ( t ). A discrete a nalogue of the bilinear defining equation for t he y n is derive d and a correspo nding represen tation of these Y n as Hank el determinan ts is given . The first few Y ablonskii-V orob’ev p o lynomials are y 0 = 1 , y 1 = t, y 2 = t 3 + 4 , y 3 = t 6 + 20 t 3 − 80 , y 4 = t 10 + 60 t 7 + 11200 t. (2) A further pro p ert y of these p olynomials, in ves tigated recen tly by Clarkson and Mansfield [9], is the distribution of t heir zero es. It is known that eac h y n ( t ) ha s no zero es in common with y n ± 1 ( t ), a nd that these zero es are simple [10 , 11]. Numerical studies indicate that these zero es lie in appro ximately tria ngular arra ys and that the zero es of y n ( t ) in terlace, in a certain sense, with those of y n +1 ( t ), in a similar w ay to the zero es of classical orthogo nal p olynomials. In section 3 w e sho w that the p olynomials Y n ( t, h ) ha v e analogous prop erties. In the contin uum case the bilinear differen tial- difference equation that defines t he Y ablonskii-V orob’ev p olynomials is give n in terms of the Hirot a deriv ative D t b y y n +1 y n − 1 = ty 2 n − 2 D 2 t y n · y n , (3) with the init ia l p olynomials y 0 = 1, y 1 = t , and this equation fo llo ws from the represen tation o f P I I as a pair of bilinear equations f or the asso ciated tau-functions. In section 4 we consider a discretisation of P I I , namely the alternate discrete P I I equation ( alt - dP I I ) studied in [12], and explain how its solutions are sp ecified by ta u-functions whic h satisfy a discrete bilinear equation together with a quadrilinear (degree four) relation. The bilinear relation defining the p olynomials Y n ( t, h ) is a consequence of these tau-function e quations. The alt - dP I I equation has B ¨ ac klund transforma t ions corresp onding to the shifts n → n ± 1, just as in the contin uum case, whic h imply R ational solutions of discr ete T o da and al t - dP I I 3 that the discrete Y ablonskii-V orob’ev p o lynomials satisfy recurrence relatio ns whic h are analogo us to those in the con tin uum setting. It is also kno wn that the alt - d P I I equation arises as the con tiguit y relations for a B¨ ac klund tra nsformation for P I I I . The latter connection leads to an alternativ e for mula for the discrete Y ablonskii-V orob’ev p olynomials in terms of determinants of Ja cobi- T rudi ty p e, corresp onding to Umemu ra p olynomials [13]. It is known from t he w ork of Ok amoto that the contin uum P I I can b e written in Hamiltonian form, and the B¨ ac klund transforma t ion is a canonical tra nsformation [14]. In section 5 we presen t a P oisson structure for the discrete case suc h that b oth dev elopmen t in time and the B¨ ac klund transforma t io n for al t - dP I I are symplectic maps. Finally in section 6 we presen t a Lax pair for al t - d P I I , whic h relat es it to the isomono dromic deforma t io n of an asso ciated linear system, and show that this tends to Jim b o and Miw a ’s Lax pair f o r the con tin uum case as h → 0. 2. Discrete time T o da latt ic e and discrete Y ablonskii-V orob’ev p olynomials The equations for the discrete T o da lattice obtained by Suris [8] are x n ( t + h ) − x n ( t ) = hπ n ( t + h ) , π n ( t + h ) − π n ( t ) = 1 h log h 1+ h 2 exp( x n − 1 ( t ) − x n ( t )) 1+ h 2 exp( x n ( t ) − x n +1 ( t )) i , (4) where π n denotes the canonically conjugate mo mentum to x n , and these discrete equations clearly yield Hamilton’s equations for the T oda lattice (1) in the con tin uum limit, as h → 0. They arise fro m the consistency condition for the Lax pair Ψ n ( t + h ) = V n ( t )Ψ n ( t ) , Ψ n +1 ( t ) = L n ( t )Ψ n ( t ) , (5) that is L n ( t + h ) V n ( t ) = V n +1 ( t ) L n ( t ), where L n ( t ) = λ exp( − hπ n ( t )) − 1 λ h exp( − x n ( t )) − h exp( x n ( t ) − hπ n ( t )) 0 ! (6) and V n ( t ) = 1 λ − h exp( − x n ( t )) h exp( x n − 1 ( t )) λ ! . (7) Up on setting Ψ n ( t ) = [ X n ( t ) , Y n ( t )] T and Z n ( t ) = X n ( t ) Y n ( t ) − 1 , w e find from the second of (5) that Z n ( t ) = − h exp( − x n ( t )) λ exp( − hπ n ( t )) − 1 λ + h exp( − hπ n ( t ) + x n ( t )) Z n +1 ( t ) . (8) This recurrence relation can b e used to generate the con tin ued fraction expansion λhZ 0 ( t ) = γ f 1 1 + γ g 1 + γ f 2 1 + γ g 2 + γ f 3 1 + γ g 3 + . . . , (9) where γ = λ 2 , f 1 = h 2 exp( − x 0 ( t )) , g 1 = − exp( − hπ 0 ( t )) and for n = 2 , 3 , . . . w e hav e f n = − h 2 exp( − x n − 1 ( t ) + x n − 2 ( t ) − hπ n − 2 ( t )) , g n = − exp( − hπ n − 1 ( t )) . (10) R ational solutions of discr ete T o da and al t - dP I I 4 Similarly fro m the first of (5) w e hav e the discrete time Riccati equation Z 0 ( t + h ) − Z 0 ( t ) = − h λ e − x 0 ( t ) +  1 λ 2 − 1  Z 0 ( t ) − h λ e x − 1 ( t ) Z 0 ( t + h ) Z 0 ( t ) . (11) In o ur previous w o rk [15] w e to ok x − 1 ( t ) → −∞ so that this b ecame a linear equation. Here w e tak e x − 1 ( t ) = 0 a nd set Z 0 ( t ) = 1 hλ W ( t ) so that with γ = λ 2 w e find W ( t + h ) = − h 2 e − x 0 ( t ) + 1 γ W ( t ) − 1 γ W ( t + h ) W ( t ) . (12) No w the con t inued fraction (9) has expansions b oth in p ositive and negativ e p ow ers of γ , namely W ( t ) = P ∞ n =1 α n ( t ) γ n and W ( t ) = P n =0 −∞ α n ( t ) γ n resp ectiv ely . Substituting these expansions in turn in to (12) and equating co efficien ts of corresp onding p o we rs of γ , w e obtain the recurrence relat io ns for their co efficien ts, giv en resp ectiv ely b y α j +1 ( t ) = α j ( t + h ) + j X k =1 α k ( t + h ) α j − k +1 ( t ) , j = 1 , 2 , 3 , . . . , (13) α j ( t + h ) = α j +1 ( t ) − 0 X k = j +1 α k ( t + h ) α j +1 − k ( t ) , j = − 1 , − 2 , . . . . (14) where α 1 ( t ) = h 2 exp( − x 0 ( t )) and α 0 ( t + h ) = − h 2 exp( − x 0 ( t )). The contin ued fraction in (9 ) is know n a s a T- f r a c tion [16] and its elemen ts are giv en in terms of Hankel determinan ts. If w e tak e the definitions u n = H ( − n +2) n , v n = H ( − n +1) n , (15) with Hank el determinan ts H ( m ) k =          β m β m +1 . . . β m + k − 1 β m +1 β m +2 . . . β m + k . . . . . . . . . β m + k − 1 β m + k . . . β m +2 k − 2          (16) whose elemen ts are give n by β k = − α k ( t ) , k = 1 , 2 , . . . , = α k ( t ) , k = 0 , − 1 , − 2 , . . . , (17) then the elemen ts of the T-fraction are give n by f n = − v n − 2 u n v n − 1 u n − 1 , g n = − v n − 1 u n v n u n − 1 , n = 2 , 3 , 4 , . . . . (18) F rom the expression for the f n , g n in terms of the co ordinates and momen ta giv en b y (10) and the equation of motion (4), it follo ws that f 1 ( t + h ) f 1 ( t ) = − g 1 ( t + h ) , g 1 ( t + h ) g 1 ( t ) = 1 + f 2 ( t ) g 1 ( t ) 1 + f 1 ( t ) (19) and more generally f n +1 ( t + h ) f n +1 ( t ) = g n +1 ( t + h ) g n ( t ) , g n +1 ( t + h ) g n +1 ( t ) = 1 + f n +2 ( t ) g n +1 ( t ) 1 + f n +1 ( t ) g n ( t ) , n ∈ N . (20) R ational solutions of discr ete T o da and al t - dP I I 5 Prop osition 2.1 The two typ es of Hankel determinant defi n e d b y ( 15) ar e r elate d by v n ( t + h ) = u n ( t ) , n = 0 , 1 , 2 , . . . . (21) Pro of: Substituting the expressions for f n , g n giv en by (18) in to the first of (20) one finds that v n +1 ( t + h ) u n ( t ) v n ( t + h ) u n +1 ( t ) = v 1 ( t + h ) u 0 ( t ) v 0 ( t + h ) u 1 ( t ) . (22) But u 0 = H (2) 0 = 1 = H (1) 0 = v 0 and u 1 ( t ) = β 1 = − h 2 exp( − x 0 ( t )), v 1 ( t ) = β 0 = − h 2 exp( − x 0 ( t − h )). Therefore the righ t ha nd side of (22) equals unity a nd the result follo ws.  Prop osition 2.2 When exp( − x 0 ( t )) = − t 4 , the Hankel determi n ants u n satisfy the biline ar r elation  1 − h 2 t 4  u n ( t + h ) u n ( t − h ) = u n ( t ) 2 + u n +1 ( t ) u n − 1 ( t ) (23) for n = 1 , 2 , . . . . Pro of: Substituting for g n and f n from (18) in to the second of (20) and then eliminating the dependence on the v n using (21), the result follows from the fact that u 1 ( t ) = v 1 ( t + h ) = h 2 t 4 and u 2 ( t ) = − h 6 64 ( t 3 − h 2 t + 4).  Remark. In [17], Hank el determinan t solutions are give n for a differen t (but gauge equiv alen t) bilinear form of the discrete T oda lattice equation, na mely the equation ρ l +1 n ρ l − 1 n − ( ρ l n ) 2 = ε 2 ρ l +1 n +1 ρ l − 1 n − 1 . Up on setting ε 2 = 1 and t = ( l − n ) h , u n ( t ) = ρ n + ht n /ρ ht 0 satisfies (23) for a suitable c hoice of the initial condition ρ ht 0 for n = 0. It follo ws from the recurrence relat io ns (14) that when exp( − x 0 ( t )) = − t 4 the α j ( t ), and hence also u n ( t ) and v n ( t ), are p olynomials in h, t . W e now renormalise the u n so that they are O ( h 0 ) as h → 0. Definition 2.1 Th e d iscr ete Y abl o nskii-V or ob’ev p o l yno m ials Y n ( t, h ) ar e define d by the biline ar r e curr enc e h 2 Y n +1 ( t, h ) Y n − 1 ( t, h ) = ( h 2 t − 4) Y n ( t + h, h ) Y n ( t − h, h ) + 4 Y n ( t, h ) 2 (24) for n ≥ 1 , with Y 0 ( t, h ) = 1 , Y 1 ( t, h ) = t as in itial da ta. Theorem 2.1 Th e discr ete Y ablonskii-V or ob ’ev p olynomials Y n ( t, h ) ∈ Z [ t, h 2 ] ar e given in terms of the Hanke l determinants u n in (15) b y Y n ( t, h ) = ( − 1) n ( − h 2 4 ) − n ( n +1) / 2 u n ( t ) for n = 0 , 1 , 2 , . . . . Ea c h Y n is a monic p olynomia l of de gr e e n ( n + 1) / 2 in t , satisfying Y n ( t, h ) = y n ( t ) + O ( h 2 ) as h → 0 , wher e the y n ( t ) ar e the usual Y ablonskii-V or ob’ e v p olynomials define d by (3) with y 0 = 1 , y 1 = t . Pro of: With α 1 ( t ) = − h 2 t/ 4 it is easy to pro ve by induction f rom the first recursion in (1 4) that α j ( t ) = h 2 P j / 4 j for j ≥ 1, where P j ∈ Z [ t, h ]. Similarly , with α 0 ( t ) = h 2 ( t − h ) / 4 the second recursion in (14) implies that α j ( t ) = h 2 ˆ P j / 4 − j +1 for j ≤ 0, where ˆ P j ∈ Z [ t, h ]. It follo ws from their definition in terms of the matrix elemen ts (17) R ational solutions of discr ete T o da and al t - dP I I 6 that the Hank el determinan ts u n are p olynomials in Q [ t, h ], with p ow ers of 4 a s the only p ossible denominators of the co efficien ts. If w e let Y n ( t, h ) = ( − 1) n ( − h 2 4 ) − n ( n +1) / 2 u n ( t ) for n ∈ N then w e find Y 0 ( t, h ) = 1, Y 1 ( t, h ) = t , and then the relatio n (24) follows fro m Prop osition 2.2, by substituting for u n ( t ) in (23) in terms of Y n ( t, h ). Since the Y n are uniquely defined b y the recurrence (24) together with the giv en initial data, the form ula in terms of renormalised Hank el determinan ts guaran t ees that they are p olynomials in t , and the recurrence also implies that they are monic and of degree d n := n ( n + 1 ) / 2) in t . Ho w ev er, further analysis is r equired to v erify that there are no p o w ers of h or p o w ers of 4 in the denominator for this c hoice of normalisation. By induction, supp ose that Y n ( t, h ) ∈ Z [ t, h ], and hence is regular as h → 0, for n = 0 , 1 , . . . , N (whic h clearly holds for N = 1). T aking a T aylor expansion in t , with deriv atives denoted b y Y N , ( j ) t = ∂ j Y N ∂ t j , w e see that Y N ( t + h, h ) Y N ( t − h, h ) =  P [ d N / 2] k =0 h 2 k Y N, (2 k ) t ( t,h ) (2 k )!  2 − h 2  P [( d N − 1) / 2] k =0 h 2 k Y N, (2 k +1) t ( t,h ) (2 k +1)!  2 = Y N ( t, h ) 2 + h 2 ˜ P ( t, h ) , where ˜ P ( t, h ) is a p olynomial in t and h by the inductiv e h yp othesis. Moreov er, at leading order we hav e ˜ P ( t, h ) = Y N ( t, 0) Y N ,tt ( t, 0) − Y N ,t ( t, 0) 2 + O ( h ) . Substituting this into the righ t hand side of (2 4 ) and dividing by h 2 w e ha v e Y N +1 ( t, h ) Y N − 1 ( t, h ) = tY N ( t + h, h ) Y N ( t − h, h ) − 4 ˜ P ( t, h ) , (25) from whic h it follows that Y N +1 ( t, h ) is r egular as h → 0 , and hence lies in Q [ t, h ] with at w orst p o w ers of 4 as denominators of its co efficien ts. No w for some K ≥ 0 w e can write Y N +1 ( t, h ) = ˆ Y N +1 ( t, h ) / 4 K where ˆ Y N +1 ∈ Z [ t, h ] with 4 6 | ˆ Y N +1 . If we m ultiply b oth sides of (24) through b y 4 K w e see that if K > 0 then w e hav e 4 | h 2 ˆ Y N +1 Y N − 1 ; but then since Y N − 1 is monic in t it is clear that 2 6 | Y N − 1 ( t, h ) w e m ust hav e 4 | ˆ Y N +1 , whic h is a con tradiction. Hence K = 0 and Y N +1 ∈ Z [ t, h ] as required. Setting h → − h in the recurrence (2 4 ) it also follows immediately b y induction that Y n ( t, h ) = Y n ( t, − h ) for all n , so in fact w e hav e p olynomials in Z [ t, h 2 ]. Th us w e can write Y n ( t, h ) = y n ( t ) + O ( h 2 ) where y 0 ( t ) = Y 0 ( t, h ) = 1 and y 1 ( t ) = Y 1 ( t, h ) = t , and using the leading order part of ˜ P ( t, h ) in (25) w e find that for n ≥ 1 the y n satisfy y n +1 y n − 1 = ty 2 n − 4 y n ¨ y n + 4 ˙ y 2 n , whic h is precisely the defining relation ( 3 ) for the usual Y ablonskii-V o rob’ev p olynomials.  The first few discrete Y ablonskii-V orob’ev p olynomials a re Y 0 ( t, h ) = 1 , Y 1 ( t, h ) = t, Y 2 ( t, h ) = t 3 + 4 − h 2 t, Y 3 ( t, h ) = t 6 + 20 t 3 − 80 + h 2 (4 t − 5 t 4 ) + 4 t 2 h 4 , R ational solutions of discr ete T o da and al t - dP I I 7 Y 4 ( t, h ) = t 10 + 60 t 7 + 11200 t − h 2 (15 t 8 + 252 t 5 + 3360 t 2 ) + + h 4 (63 t 6 + 480 t 3 + 576) − h 6 (85 t 4 + 288 t ) + 36 h 8 t 2 . (26) These examples clearly reduce to the usual Y ablonskii-V orob’ev po lynomials (2 ) when h = 0. 3. Zero es of discrete Y ablonskii-V orob’ev p olynomials In the con tin uum case it has b een show n t hat the zero es of y n ( t ) are simple a nd are not zero es of y n +1 ( t ) [10, 11]. Here w e pro ve an analogous result in the discrete case, b y considering the ro ots of Y n ( t, h ) as a p olynomial in t , that is t 0 = t 0 ( h ) suc h that Y n ( t 0 ( h ) , h ) = 0. T o b egin with we require a simple observ ation. Lemma 3.1 Th e d iscr ete Y ablonsk i i-V or ob’ev p olynomial s n ever vanish at t = 4 /h 2 . Mor e pr e cisely, Y n  4 h 2 , h  =  4 h 2  n ( n +1) / 2 6 = 0 f or all n ∈ N . Pro of: The result is trivially true when n = 0 , 1. Assume that it holds for n = 1 , 2 , . . . , N . Then from (24) h 2 Y N +1 (4 /h 2 , h ) Y N − 1 (4 /h 2 , h ) = 4 Y N (4 /h 2 , h ) 2 6 = 0, and the result follows by induction.  F ro m Theorem 2.1 w e kno w that Y n ( t + h, h ) − Y n ( t, h ) = h ˙ y n ( t ) + O ( h 2 ), and the zero es o f Y n ( t, h ) in t differ f rom those of y n ( t ) by O ( h 2 ), so (b ecause the zero es of y n are simple) w e expect that Y n ( t + h, h ) should not v anish where Y n ( t, h ) do es. Indeed this turns o ut t o b e the case. In the next section w e shall see that the p olynomials Y N are tau-functions for a discrete PI I equation, and hence satisfy man y other bilinear iden tities in addition to (24). T o prov e the follow ing result w e mak e use of o ne such iden tity , namely (50) b elow . Theorem 3.1 F or any n = 1 , 2 , . . . , if t 0 is a zer o of Y n , so that Y n ( t 0 , h ) = 0 , then Y n ( t 0 ± h, h ) 6 = 0 and Y n +1 ( t 0 , h ) 6 = 0 . Pro of: It is true for n = 1 by insp ection. Suppo se the result is true fo r n = 1 , 2 , . . . , N − 1. Now assume that b oth Y N ( t 0 , h ) = 0 a nd Y N ( t 0 + h, h ) = 0. Using the iden tity (50) with τ n = Y n ( t, h ), and setting n = N − 1, t = t 0 , w e hav e Y N ( t 0 + h, h ) Y N − 2 ( t 0 , h ) − Y N ( t 0 , h ) Y N − 2 ( t 0 + h, h ) = (2 N − 1) hY N − 1 ( t 0 + h, h ) Y N − 1 ( t 0 , h ) . By the assumption, the left hand side v anishes, and 2 N − 1 6 = 0 for N ∈ Z whic h means that Y N − 1 ( t 0 , h ) or Y N − 1 ( t 0 + h, h ) = 0; but this con tradicts the inductiv e h yp othesis, so Y N ( t 0 + h, h ) 6 = 0 whene v er Y N ( t 0 , h ) = 0 (and similarly for Y N ( t 0 − h, h )). No w if Y N ( t 0 , h ) = 0, t hen from (24) w e see that h 2 Y N − 1 ( t 0 , h ) Y N +1 ( t 0 , h ) = ( h 2 t 0 − 4) Y N ( t 0 + h, h ) Y N ( t 0 − h, h ) R ational solutions of discr ete T o da and al t - dP I I 8 If w e no w a ssume t hat also Y N +1 ( t 0 , h ) = 0, then the left hand side of the ab o ve v anishes, and since h 2 t 0 − 4 6 = 0 by Lemma 3.1 then Y N ( t 0 + h, h ) = 0 or Y N ( t 0 − h, h ) = 0, which is a con tradiction. Therefore the result holds for n = N , hence for all n by induction.  Clarkson and Mansfield hav e n umerically studied the lo cation of zero es of y n ( t ) for lo w v alues of n [9]. They found that for a giv en n they o ccup y a ppro ximate triangular arra ys in t he complex plane a nd that the zero es o f y n ( t ) i nterlac e in a certain sense with those o f y n +1 ( t ). Since the zero es o f Y n ( t, h ) are the same as those of y n ( t ) up to O ( h 2 ), the same picture holds for sufficien tly small h , and our n umerical observ ations sho w that the same qualitative b ehaviour p ersists for v alues o f h up to at least or der of unit y . 4. Alternate discrete Pain lev´ e I I The second P ainlev ´ e equation ( P I I ) is ¨ q = 2 q 3 + tq + α, (27) where α is a constan t. This second order differen tial equation is equiv alen t to a first order system in tro duced b y O k amo t o [1 4], namely ˙ q = p − q 2 − t 2 ˙ p = 2 q p + ℓ, ℓ = α + 1 2 . (28) Rational solutions of (27 ) arise when α = n ∈ Z . Ho w ev er, in this and subsequen t sections n need not b e an integer unless stated explicitly otherwise, and w e use the parameters n and α in t erchangeably . W e also find it con venie n t to use t he shifted parameter ℓ = n + 1 / 2 as ab ov e, whic h fixes a p oint in the A 1 ro ot space. This is the pa rameter used by Ok amoto to lab el solutions of P I I and the cor r espo nding tau- functions, but w e stic k with the lab el n to main tain con tact with the preceding results. P I I ma y b e put into bilinear form b y setting q n = d dt log  τ n − 1 ( t ) τ n ( t )  . (29) Then(27) with α = n is equiv alen t to the bilinear equations D 2 t τ n · τ n − 1 = F ( t ) τ n τ n − 1 , ( D 3 t − tD t + n ) τ n · τ n − 1 = 3 F ( t ) D t τ n · τ n − 1 , (30) where D t is the usual Hirota o p erator, and the function F ( t ) is a rbitrary . By a g auge transformation on t he tau-functions, that is τ n → G ( t ) τ n for all n , the function F can b e set to zero in (30) without loss of generalit y . The B¨ ac klund tra nsformation linking a solution o f (27) for α = n with that for α = n + 1 is q n +1 = − q n − ℓ p n , p n = ˙ q n + q 2 n + t/ 2 , (31) R ational solutions of discr ete T o da and al t - dP I I 9 and the tw o solutions are related b y ˙ q n + q 2 n = − ˙ q n +1 + q 2 n +1 (32) (whic h can b e seen as b eing inherited fro m the Miura tra nsfor ma t io n for KdV). F rom (30) a nd (31) it ma y b e pro ved that, with the same c hoice of gauge that fixes F ≡ 0 in (30), an y three adj a cen t t a u-functions τ n , τ n ± 1 satisfy the bilinear relations D t τ n +1 · τ n − 1 = 2 ℓ τ 2 n (33) and τ n +1 τ n − 1 = tτ 2 n − 2 D 2 t τ n · τ n . (34) With this ch oice of gauge, the equation (34) is the bilinear form of the T o da lattice. As w e hav e alr eady men tioned (cf. equation (3) in the Intro duction), the latter relation is the defining recurrence fo r the Y ablo nksii-V orob’ev p olynomials; with τ n = y n they satisfy the ot her bilinear equations (33) and (30) fo r F = 0, and hence determine the rational solutions of P I I . Sev eral discretisations of P I I ha v e b een studied in recen t y ears including a discrete bilinear fo rm [18] whic h tends to (3 0 ) in the contin uum limit. The discrete bilinear form allow ed the construction of rational solutions to this discrete P I I , giv en in terms of ratios o f determinan t s of Jacobi-T rudi t ype [19], while solutions in terms of discrete Airy functions w ere found in [20, 21]. A hierarc hy o f higher o r der analo g ues o f this discrete P I I ( dP I I ) ha v e b een giv en by Cressw ell and Joshi, based on a Lax pair. How ev er, due to the non-uniqueness o f t he discretisation pro cess , there are ma ny other dP I I equations with a na logous prop erties. F o r example, four differen t dP I I equations are men tioned in [23], three of whic h ar e q - t yp e ( q - P I I ). Discrete Airy function solutions of o ne suc h q - P I I equation are f o und in [24], while another one is considered with its ultra- discretisation in [25], along with y et one more distinct q - P I I . In [26] a detailed study of a q - P I I in Sak a i’s classification [27] ha s b een p erformed. Another a lt ernativ e discretisation, whic h is relev an t here, is the so called a lterna te dP I I ( alt - dP I I ) whic h was studied in [1 2], and app eared in connection with the orig ina l d P I I in [28]. Here w e consider the alt - d P I I equation in the form of t he second order difference equation h 3 2 m − 2 g g + 1 + h 3 2 ( m − 1) − 2 g g + 1 = − g + 1 g + h 3 2 ( m − ℓ ) − 2 (35) with the notation g = g n ( t ), g n = g n ( t + h, h ), g n = g n ( t − h, h ) ‡ . Th is is referred to as a discrete P I I equation due to the fact that (2 7) arises fro m it b y setting g ( t, h ) = − 1+ hq ( t, h ) and taking the con tinuum limit h → 0 (with parameter α = ℓ − 1 / 2 as usual). If w e set g n ( t, h ) = x m with z m = h 3 m/ 2 − 2 and µ = − h 3 ℓ/ 2 then t his is equation (1.3) in [12 ] (except that we hav e m in place of n ). ‡ The in tro duction of m in place of t is not ess ent ial, but it sugges ts that m and ℓ might be co nsidered on the s ame fo o ting , so that w ℓ,m ≡ g n ( t ) should pr ovide particular solutions of a suitable partia l difference equa tion. See [12] for a connection with the lattice mo dified BSQ equation. R ational solutions of discr ete T o da and al t - dP I I 10 The al t - dP I I equation (35) has ratio na l solutions for inte ger v alues of the parameter n = ℓ − 1 / 2, pro vided by suitable ratios of the discrete Y a blonskii-V orob’ev p olynomials Y n ( t, h ). In order to prov e this, w e must deriv e appropria te discrete analogues of the bilinear iden tities (3 0), (33) and (34). W e star t b y presen ting (35) in the for m of a system, whic h mak es it more manag eable. Lemma 4.1 Th e al t - dP I I e quation (35) for ℓ = n + 1 2 , m = t/h is e quivalen t to the first or der system g n = ( − 2 + h 2 p n ) / (4 + 2 g n − h 2 ( g n p n + t )) , p n = ( p n − ℓhg n ) /g 2 n . (36) Pro of: By rearranging the first equation in (36 ) w e ha v e p n = 1 h 2  (4 − h 2 t ) g n 1 + g n g n + 2  , (37) and by shifting t → t + h the latter pro vides a formula for p n in terms of g n and g n , and then substituting this in to the left hand side o f the second equation (36), with p n giv en b y (37) on the r ig h t hand side, gives a relation b et w een g n , g n and g n . After rearra nging, shifting t → t − h , and setting t = mh , this relation is precisely (35 ). Con v ersely , giv en the al t - d P I I equation (35), w e can define p n b y the formula (37), whic h is equiv alen t to the first equation in (36); the second equation for p n then follows immediately from alt - d P I I .  The B¨ ac klund tr ansformation for the al t - dP I I equation (35 ) is most easily deriv ed starting from the analog ue of (32), and the following results are easily v erified b y direct calculation. Lemma 4.2 I f g n +1 is given by g n +1 = p n ℓh + g n p n , (38) in terms of g n and p n satisfying (36), then the id entity 1 g n + g n = 1 g n +1 + g n +1 (39) holds. Corollary 4.1 The quantity g n +1 , define d by (38) with p n given by (37) , satisfies (35) with ℓ → ℓ + 1 . Equivalently, the e quation (38) an d the r elation p n +1 = 1 h 2  (4 − h 2 t ) g n +1 1 + g n +1 g n +1 + 2  (40) to g ether c onstitute a B¨ acklund tr an sformation for the al t - d P I I system (36). Corollary 4.2 The B ¨ acklund tr ansformation for the al t - dP I I system, given by the formulae (38) and (40), has the fol lowing c onse quenc es: g n +1 p n = g n p n ; (41) R ational solutions of discr ete T o da and al t - dP I I 11 1 g n − g n +1 = ℓh p n ; (42) g n +1 g n − 1 g n +1 g n = − ℓh p n  g n + 1 g n  . (43) Remark. With g n = − 1 + hq n , the iden tit y (32) arises as the con tinuum limit o f (39), and the formula (31) arises from (38), as h → 0. Similarly , the system (28) is the con tinuum limit of (3 6). Nijhoff et al. deriv ed equiv alen t formulae of Miura/Sch lesinger t ype fo r the B¨ ac klund transforma t ion of al t - dP I I b y making use of a v ariable y n (see equation (5.1) in [12]), whic h (mo dulo rescaling and replacing n by m ) is a nalogous to p n defined b y (37). W e no w describe the t a u-functions for the alt - d P I I equation, whic h satisfy analogues of (30). Prop osition 4.1 Up to a choic e of gauge, every solution of (35) is sp e cifie d by a p air of tau-functions τ n ( t, h ) , τ n − 1 ( t, h ) via the formula g n ( t, h ) = − τ n − 1 ( t − h, h ) τ n ( t, h ) τ n − 1 ( t, h ) τ n ( t − h, h ) , (44) wher e the tau-functions satisfy the bilin e ar e quation τ n τ n − 1 + τ n τ n − 1 = 2 τ n τ n − 1 (45) and the quadriline ar (de gr e e four) e quation (4 − mh 3 ) τ n − 1 τ n ( τ n − 1 τ n − τ n − 1 τ n ) + (4 − ( m − 1) h 3 ) τ n − 1 τ n ( τ n − 1 τ n − τ n − 1 τ n ) +8( τ 2 n − 1 τ 2 n − τ 2 n − 1 τ 2 n ) − 4 nh 3 τ n − 1 τ n τ n − 1 τ n = 0 , (46) with m = t/h , n = α = ℓ − 1 / 2 and τ n = τ n ( t, h ) , τ n = τ n ( t + h, h ) , τ n = τ n ( t − h, h ) , etc. Pro of: Up on substituting the tau-function expres sion ( 4 4) in to (35) and clearing denominators, a relation of degree eigh t results, whic h can b e simplified somewhat b y rewriting it in terms of the symmetric/an tisymmetric quadratic quan tities A ± = τ n − 1 τ n ± τ n − 1 τ n and A ± = τ n − 1 τ n ± τ n − 1 τ n . In general, for a ny choice of tau-functions the bilinear equation τ n τ n − 1 + τ n τ n − 1 = 2 ˆ F τ n τ n − 1 holds, for some f unction ˆ F = ˆ F ( t, h ), but b y applying a gauge transformation τ n → ˆ G τ n , τ n − 1 → ˆ G τ n − 1 with ˆ G ˆ G/ ˆ G 2 = ˆ F the function ˆ F can b e remo v ed t o yield the bilinear equation (45). With this c hoice of gauge, t he remaining terms in the degree eight relation factorise to yield the quadrilinear equation (46), and conv ersely if these t wo tau-function equations hold then g n giv en b y (44) is a solution of (35) fo r ℓ = n + 1 / 2 = α + 1 / 2.  Remark. The existence of a quadrilinear relation b etw een a pair of tau-functions is men tioned in section 4 of [1 2], where a third tau-function is introduced to obtain purely bilinear relations (cf. Theorem 4.1 b elo w). R ational solutions of discr ete T o da and al t - dP I I 12 It is easy to see tha t (45) tends to the first of (30) in the con tin uum limit (with the g auge c hosen so that F = 0). Although the second relation (46) b et w een the t wo tau-functions is of o v erall degree f o ur, it still pro duces the second bilinear differen tia l equation (30) in the con tinuum limit (provided that the first one also holds). In order to w ork with purely bilinear equations in the discrete case, w e m ust consider three adjacent tau-functions τ n , τ n ± 1 . Theorem 4.1 Up to a choic e of gauge, every solution of the al t - dP I I system ( 3 6) is sp e cifie d by thr e e tau-functions τ n − 1 ( t ) , τ n ( t ) , τ n +1 ( t ) , with g n given by (44) and p n = τ n − 1 τ n +1 2 τ 2 n , (47) wher e the tau-functions satisfy (45) as wel l as τ n +1 τ n + τ n +1 τ n = 2 τ n +1 τ n (48) and h 2 τ n +1 τ n − 1 = ( h 2 t − 4) τ n τ n + 4 τ 2 n . (49) With this choic e of norm a lisation the ide n tities τ n +1 τ n − 1 − τ n +1 τ n − 1 = 2 ℓhτ n τ n (50) and τ n +1 τ n − 1 − τ n +1 τ n − 1 = 4 ℓhτ 2 n (51) also h o ld. F or this cho ic e of g a uge , these pur ely biline ar r e lations ar e c omp atible wi th the B¨ acklund tr ansformation (38) for al t - dP I I , in the sense that τ n and τ n +1 satisfy (46) for n → n + 1 , and g n +1 ( t, h ) = − τ n ( t − h, h ) τ n +1 ( t, h ) τ n ( t, h ) τ n +1 ( t − h, h ) satisfies (35) with ℓ → ℓ + 1 . Pro of: If a solution g n , p n of (36) is giv en b y the expressions (44) and (47) resp ectiv ely , and the gauge is fixed b y (45), then the latter implies that g n + 1 g n = − 2 τ 2 n τ n τ n = 1 g n +1 + g n +1 b y Lemma 4.2, where g n +1 giv en b y (38) is a solution of (35) with ℓ → ℓ + 1. The first equalit y ab ov e implies (49), while the relation (41) implies that g n +1 is giv en in terms of tau-functions b y the form ula (44) with n → n + 1, and hence ( 4 8) follows from the second equalit y a b o v e. The bilinear iden tities (50) and ( 5 1) then hold a s a consequence of the relat io ns (42) and ( 4 3) resp ectiv ely . By Prop osition 4.1, the giv en c hoice of g auge implies that the pair τ n − 1 , τ n satisfy (46), and the fact that g n +1 is a solution of al t - d P I I with the pa rameter shifted implies t ha t the pair τ n , τ n +1 also satisfy this quadrilinear equation with n → n + 1. R ational solutions of discr ete T o da and al t - dP I I 13 Con vers ely , supp ose that g n is defined in terms of tau-functions b y (4 4 ), g n +1 is defined b y the same relation for n → n + 1, and p n is defined b y (47), where the tau- functions satisfy the t hree relatio ns (45), (48) and (49). The iden tities (37) and (39) follo w immediately . F urthermore, fro m ( 45) and (48) it is clear that τ n − 1 τ n  τ n +1 τ n + τ n +1 τ n  = τ n +1 τ n  τ n − 1 τ n + τ n − 1 τ n  , whic h implies that τ n +1 τ n − 1 − τ n +1 τ n − 1 τ n τ n = τ n +1 τ n − 1 − τ n +1 τ n − 1 τ n τ n . Therefore ( τ n +1 τ n − 1 − τ n +1 τ n − 1 ) / (2 hτ n τ n ) is indep enden t o f t , and if w e denote this by ℓ , then we ha v e the bilinear equation (50), whic h implies that (4 2) a lso holds. Solving (42) for g n +1 giv es an expression in terms of p n and g n , whic h in turn means that g n +1 can b e written in terms of g n and g n using ( 3 7). By shifting t → t − h , this giv es a form ula for g n +1 in terms of g n and g n , and then substituting f or g n +1 and g n +1 in (39) yields the al t - dP I I equation (35). It then follows t ha t g n , p n satisfy the system (36).  Theorem 4.2 F or p ar ameter ℓ = n + 1 2 with n ∈ Z the alt - dP I I e quation (35) has r ational solutions given in terms of the discr ete Y ablons k ii-V or ob’ev p olynomials by g n = − Y n − 1 ( t − h, h ) Y n ( t, h ) Y n − 1 ( t, h ) Y n ( t − h, h ) , wher e the p o l ynom ials ar e extende d to ne gative indic es n by s e tting Y − n = Y n − 1 for n ∈ N . As wel l as the defining r e curr enc e (24), the r elations (45), (46), (50) and (5 1) ar e satisfie d by τ n ( t, h ) = Y n ( t, h ) for al l n ∈ Z . Pro of: When ℓ = 1 / 2, the equation (35) has the trivial constan t solution g 0 ( t, h ) = − 1, whic h can b e obtained b y setting τ 0 = Y 0 = 1 = Y − 1 = τ − 1 in (44), and f r o m (37) w e ha v e p 0 = t/ 2 whic h give s τ 1 = Y 1 = t b y (47). It is easy to v erify that eac h of the bilinear equations (45), (48) and (49) is satisfied by these tau-functions. By applying the B¨ ac klund transformation (38) rep eatedly ( b oth forwards and backw a rds) a doubly infinite sequence of rationa l solutions { g n } n ∈ Z is obtained. Then b y Theorem 4 .1, since the B¨ ac klund transformation is compatible with the c hoice of gauge, it follow s b y induction that the corresp onding tau-functions satisfy the iden tities (24), (45), (46), (50) and (51) for a ll n ∈ Z . Since the Y ablonskii-V oro b’ev p olynomials are defined by (49) with τ 0 = 1, τ 1 = t , it follows that this pa rticular sequence of tau- functions is giv en b y τ n ( t, h ) = Y n ( t, h ) for all n ∈ N . The fact that this relation can b e consisten tly extended to negativ e n follo ws from the observ ation tha t all of the tau-function iden tities in Prop osition 4.1 and Theorem 4.1 a re inv ariant under n → − n − 1 , ℓ → − ℓ .  Remark. The simplest r a tional solutions of alt - d P I I (corresp onding to n = 0 , ± 1) are describ ed in section 6 of the pap er [12] by Nijhoff et al. , where it is indicated how the ab o v e sequence o f rational solutions can b e generated recursiv ely via the B¨ ack lund transformation, but no closed form for these r ational solutions is giv en in that w o rk. R ational solutions of discr ete T o da and al t - dP I I 14 The fact that the al t - dP I I equation can b e deriv ed from a sequence o f B¨ ac klund transformations applied to solutions of P I I I pro vides a relation b et w een the discrete Y ablonskii-V orob’ev p olynomials and the Umem ura p olynomials for P I I I . The third P ainlev´ e equation, P I I I , is giv en by w ′′ = ( w ′ ) 2 w − w ′ x + 1 x ( αw 2 + β ) + γ w 3 + δ w , (52) with the prime ′ denoting d/dx , and without loss of generalit y (by rescaling the indep enden t v ar iable x ) if γ δ 6 = 0 the latter tw o parameters can b e fixed as γ = − δ = 1. (Note that the parameter α in (52) should not b e confused with the parameter α in P I I .) B¨ ac klund tra nsformations for P I I I (cf. section 2 in [1 2]) can b e used to relate three adjacen t solutions w n = w ( x ; α , β ) a nd w n ± 1 = w ( x ; α ± 2 , β ± 2) and if we set g n = − 1 /w n then this con tiguit y relation can b e written in the form of the al t - dP I I equation z n g n +1 g n − 1 + z n − 1 g n g n − 1 − 1 + x 2  g n + 1 g n  + z n + µ = 0 , (53) where z n = ( α + β + 2) / 4 , ∆ z n := z n +1 − z n = 1 , µ = ( β − α − 2) / 4 . (54) (T o compare with equation (2.5) in [1 2], set g n = ix n , x = t in the ab ov e.) F or a certain set of parameter v a lues, P I I I has rat io nal solutions whic h are describ ed by the follow ing result. Theorem 4.3 (Ka jiwar a & Mas uda [13]) F or p ar ameters α = 2 n + 2 ν − 1 , β = 2 n − 2 ν + 1 , n ∈ Z (55) and γ = − δ = 1 , the thir d Painle v´ e e quation (52) has r ational solutions w ( x ; α, β ) = w n given by w n = D n ( x, ν − 1) D n − 1 ( x, ν ) D n ( x, ν ) D n − 1 ( x, ν − 1) (56) wher e the p olynomial D n is given by a determi n ant of Jac ob i - T rudi typ e, D n ( x, ν ) =          p n p n +1 . . . p 2 n − 1 p n − 2 p n − 1 . . . p 2 n − 3 . . . . . . . . . . . . p − n +2 p − n +3 . . . p 1          with p k = p k ( x, ν ) defin e d by the gen e r ating function ∞ X k =0 p k ( x, ν ) λ k = (1 + λ ) ν exp( xλ ) and p k = 0 for k < 0 . R ational solutions of discr ete T o da and al t - dP I I 15 Remarks. The polynomials p k (( x, ν ) are essen tially just associated Laguerre p olynomials, and, for eac h n , D n ( x, ν ) is a Sc h ur p olynomial with r estricted argumen ts, corresp onding to the partitio n ( n, n − 1 , . . . , 2 , 1). The result stated ab o ve is an adapted form of Theorem 1 in [1 3], and describes one fa mily of rational solutions of P I I I ; for a complete description of all ratio na l solutions of P I I I for γ δ 6 = 0, see [29 ]. The p olynomials D n ( x, ν ) (after some scaling) are kno wn as the Umem ura p olynomials for P I I I . F urther prop erties of scaled Umem ura p o lynomials for P I I I are detailed in [30], including the remark a ble patterns formed b y t he ro ots, and differen tial/difference equations; analog ous po lynomials corresp onding to the sp ecial cases when γ δ = 0 are also treated there. Theorem 4.4 Th e discr ete Y ablonskii-V or ob’ e v p olynom i als ar e given in terms of determinants of Jac obi-T rudi typ e by the formula Y n ( t, h ) = c n h n ( n +1) / 2 D n  4 h 3 , t h − 4 h 3  , (57) wher e c n = (2 n − 1)!!(2 n − 3)!! . . . 3!!1!! (58) for n ∈ N . The pro o f of the preceding theorem mak es use of some results in the next section, and is relegated to the app endix. How eve r, it is clear that if w e rearrange the form ula (57) then w e can rewrite D n ( x, ν ) in the f orm of a Hank el determinan t. Corollary 4.3 The Umemur a p olynomials for P I I I , given in sc ale d form b y D n ( x, ν ) , ar e pr op ortional to the Hankel determinants u n as in (15) with h = ( x/ 4) − 1 / 3 and t = ( x/ 4) − 1 / 3 ( x + ν ) . Remark. It is kno wn that the P a inlev ´ e differen tial equations form a coalescence cascade from P V I do wn to P I (see [31]). In [13] it is sho wn that the coalescence limit from P I I I to P I I pro duces t he Y ablonskii-V oro b’ev p olynomials y n ( t ) as a limit of the Umem ura p olynomials, but this arises in a differen t w ay compared with the limit h → 0 considered ab ov e. More precisely , with the scaling used here, the coalescence from ( 5 2) to (27) a rises when the indep enden t v ariable x a nd pa rameter ν scale as x = t ǫ + 4 ǫ 3 , ν = 1 2 − 4 ǫ 3 , with ǫ → 0. In this limit, up to scaling the p o lynomials D n ( t/ǫ + 4 /ǫ 3 , 1 / 2 − 4 /ǫ 3 ) pro duce y n ( t ) at leading order in ǫ . 5. Symplectic prop erties and discrete P X X X I V Ok a moto [14] sho w ed that P I I can b e written as t he system (28) whic h is in Hamiltonian form, i.e. ˙ q = ∂ H ∂ p , ˙ p = − ∂ H ∂ q , (59) R ational solutions of discr ete T o da and al t - dP I I 16 with H = p 2 2 − ( q 2 + t 2 ) p − ℓq . (60) Eliminating p gives (2 7), whic h is P I I , whilst eliminating q give s ¨ p = ˙ p 2 2 p + 2 p 2 − tp − ℓ 2 2 p (61) whic h is kno wn as P X X X I V (see [31], c hapter XIV). This represen ta t ion has b een used in [32] to study further prop erties of the Y ablonskii-V orob’ev p o lnomials. Although the alt - d P I I equation (35), b eing a non- autonomous difference equation, do es not hav e a Hamiltonian form, man y of the results pro v ed there ha v e their counterparts in the discrete case. F or ease of comparison, w e briefly recall some kno wn results on P I I . In terms of t he canonical co ordinates ( q n , p n ), the B¨ ac klund transformation for P I I can b e written (t o gether with its inv erse) as q n +1 = − q n − ℓ p n , p n +1 = − p n + t + 2  q n + ℓ p n  2 ; q n − 1 = − q n − ( ℓ − 1) 2 q 2 n − p n + t , p n − 1 = − p n + t + 2 q 2 n . (62) It is straightforw ard to c heck that dq n +1 ∧ dp n +1 + dH n +1 ∧ dt = dq n ∧ dp n + dH n ∧ dt , so the transformation n → n + 1 is a canonical transformation on the extended phase space with co ordinates ( q n , p n , t ). The generating function for this canonical transformation is F ( q n , q n +1 ) = ℓ log( q n +1 + q n ) + 2 3 q 3 n +1 + tq n +1 (63) so that p n = − ∂ F ∂ q n , p n +1 = ∂ F ∂ q n +1 . The B¨ ac klund tr ansformation formulae (62) imply that a n y sequence o f solutions q n of P I I (lab elled b y n = ℓ − 1 / 2) satisfies ℓ q n +1 + q n + ℓ − 1 q n + q n − 1 + 2 q 2 n + t = 0 , (64) whic h is a discrete form of P I , whilst p n satisfies ( p n +1 − p n − 1 ) 2 p 4 n − 4 ℓ 2 ( p n +1 + 2 p n + p n − 1 − 2 t ) p 2 n + 4 ℓ 4 = 0 . (65) W e will now sho w ho w the ab ov e results car r y o v er in to the discrete case. Prop osition 5.1 In terms of the variables q n = ( g n + 1) /h and p n , the B¨ acklund tr an sformation (38) for the alt - d P I I e quation (35) (c orr esp onding to n → n + 1 ) c an b e written to gether with its inve rs e (c orr e s p onding to n → n − 1 ) as q n +1 = q n p n + ℓ h ( q n p n + ℓ ) − p n , p n +1 = ( q n p n + ℓ ) 2  2 p 2 n − h 2 p n  − h p n ( t − 2 p n )( q n p n + ℓ ) − p n + t ; q n − 1 = − q n + 1 − ℓ + hq n ( − 2 q 3 n + q n p n − q n t + ℓ − 1)+ h 2 q 3 n ( − 2 p n + t )+ h 3 q 4 n p n − p n + t +2 q 3 n − hq n (2 q 2 n − 3 p n +2 t )+ h 2 q 2 n ( − 3 p n + t )+ h 3 q 3 n p n , p n − 1 = − p n + t + 2 q 2 n + h (2 p n − t ) q n − h 2 p n q 2 n . (66) R ational solutions of discr ete T o da and al t - dP I I 17 This B¨ acklund tr ansformation is a symple ctic m ap in the phase sp ac e with c o or din ates ( q n , p n ) , w ith gener ating function F ( q n , q n +1 ) = 2 q n +1 h 2 + 2 h 3 (1 − hq n +1 ) − 1 h 3  h 2 t − 4 + ℓh 3  log(1 − hq n +1 ) + ℓ log ( hq n q n +1 − q n +1 − q n ) (67) such that d F = p n +1 dq n +1 − p n dq n ; in other wor ds the c anonic al Poisson br acket { q n , p n } = 1 is pr ese rv e d for n → n + 1 . The e quations (66) and the gener ating function r e duc e to those of the B¨ acklund tr ans f o rmation for P I I in the c on tinuum limit h → 0 . Pro of: The form ulae (66) follo w immediately from Lemma 4.2 and Corollary 4.1. T o v erify tha t the map ( q n , p n ) 7→ ( q n +1 , p n +1 ) is symplectic, it is sufficien t to calculate directly that its Jacobian determinan t is equal to 1. This a lso follows directly fro m the closure of the exact one-form d F , up on chec king that p n = − ∂ F ∂ q n , p n +1 = ∂ F ∂ q n +1 with the generating function F as in the form ula (67 ). It is straigh t f orw a r d to v erify that the relations (66) ha ve the correct con tin uum limit giv en b y (62), and that (67) also yields (63) when h → 0.  Remark. The symplectic structure f or the al t - dP I I equation can b e deriv ed from Ok a moto’s Hamiltonian formulation for P I I I , since the canonical co ordinates ( q n , p n ) ab ov e are related to those for P I I I b y a shift and rescaling (cf. section 2.3 in [30] for instance). Corollary 5.1 F or the al t - dP I I e quation (35), the analo g ue of (64) i s the e q uation ℓ q n +1 + q n − hq n +1 q n + ( ℓ − 1) q n + q n − 1 − hq n q n − 1 + 2 q 2 n 1 − hq n + t − hℓ = 0 . (68) In terms of g n this is the se c ond or der diffe r enc e e quation ℓ g n +1 g n − 1 + ( ℓ − 1) g n g n − 1 − 1 + 2 h 3  g n + 1 g n  + ℓ + 4 h 3 − t h = 0 , (69) which is an other form of the alt - dP I I e quation. The c o njugate momentum p n satisfies a thir d or der r e curr enc e r elation in n , na m ely p n +1 { p n 2 n +1 − hp n [(2 n − 1) 2 +2 p 2 n − 1 ( p n − p n − 2 )] 4 p n − 1 (4 n 2 − 1) − h 2 p n t 4(2 n +1) + h 3 p n (2 n − 1) 8(2 n +1) } + p n − 2 {− p n − 1 2 n − 1 − hp n − 1 [(2 n +1) 2 +2 p 2 n ( p n − 1 − p n +1 )] 4 p n (4 n 2 − 1) + h 2 p n − 1 t 4(2 n − 1) + h 3 p n − 1 (2 n +1) 8(2 n − 1) } + h 2 p 2 n p 2 n − 1 2(4 n 2 − 1) + h (4 n 2 − 1) 8 p n p n − 1 − (2 n +1)[ h 3 (2 n − 1) − 2 h 2 t +8] 16 p n − (2 n − 1)[ h 3 (2 n +1)+2 h 2 t − 8] 16 p n − 1 + p n p n − 1 [ 8 − 2 h 2 t +(4 n 2 − 3) h 3 4(4 n 2 − 1) ] − (6 n +5) hp n 4(2 n +1) − (6 n − 5) hp n − 1 4(2 n − 1) + h 32 [32 t + 16 h − 4 h 2 t 2 − 4 h 3 t + (4 n 2 − 1) h 4 ] = 0 . (70) Pro of: Up on solving the first of (66) for p n and substituting in the last o ne, (68) results. One can eliminate the quadratic terms in q n from the second and fourth of (66) to get q n = − 2 n + 1 4 p n − 2( p n +1 − p n − 1 ) p n + ( t − 2 p n )(2 n + 1) h 2(2 n + 1)( h 2 p n − 2) (71) After substituting this in to the first of (6 6) with n → n − 1 , w e obtain (70). R ational solutions of discr ete T o da and al t - dP I I 18 Remarks. The fact that the al t - dP I I equation is self-dual, in the sense that the sup erp osition formula (69) f or its B¨ ac kund transformation is (up to r escaling and rev ersing the roles of the dep enden t v ariable a nd the B¨ ac klund par a meter) the equation itself, w as first noted in [12]. Equation (37) in [32] is the con tinuum limit of equation (70). The la tter relation allo ws one to o btain p n +1 uniquely give n p n , p n − 1 , p n − 2 . Remem b ering t hat p n = τ n − 1 τ n +1 2 τ 2 n , w e see that if τ j ( t, h ) for j = n − 3 , n − 2 , . . . , n + 1 are giv en for a particular v alue of t , then τ n +2 can b e ev aluated for t his same v alue of t . More precisely , (70) is equiv alen t to a recurrence relatio n for the τ n that in volv es no shifts in t and is linear in τ n +2 and τ n − 3 ; in particular this relation, whic h is of fifth order in n , is satisfied by the discrete Y ablonskii-V orob’ev p olynomials Y n for n ∈ Z . The latter recurrence fo r the ta u- functions, whic h is omitted here, tends to equation (38) in [32] in the con tin uum limit. It is also p ossible to write q n in terms of unshifted τ n , b y substituting the rig h t hand side o f (47) for p n in (71). W e no w consider the map in the ( q n , p n ) phase space corresp onding to shifting in t rather than n . Prop osition 5.2 In the phase sp ac e with c o o r d i n ates ( q n , p n ) , the shift t → t + h c orr esp onding to the al t - dP I I system (36) is given by q n = − 2 q n + h ( t − 2 p n )+ h 2 p n q n − 2 − 2 hq n + h 2 ( t − p n )+ h 3 q n p n , p n = 1 ( h 2 p n − 2) 2  − 2 − 2 hq n + h 2 ( t − p n ) + h 3 q n p n  × ×  − 2 p n − 2 h ( ℓ + q n p n ) + h 2 p n ( t − p n ) + h 3 p n ( ℓ + q n p n )  . (72) This is a symple ctic map with the gener a ting function S = 1 h 3 (4 − h 2 t ) log(2 − h 2 p n ) + ℓ 2 log p n p n + 1 h  p n − p n  + 1 h p ℓ 2 h 2 + 4 p n p n − ℓ tanh − 1  √ ℓ 2 h 2 +4 p n p n ℓh  (73) such that dS = q n dp n − q n dp n . Pro of: The equations (72) just corresp o nd to rewriting (36) in terms o f q n = ( g n + 1) / h , q n = ( g n + 1) / h . It is extremely easy to chec k directly from (36) that the symplec tic form ω n = dq n ∧ dp n = 1 h dg n ∧ dp n is preserv ed b y t he shift in t . T o find the generating function, it is conv enien t to write g n , g n in terms of p n , p n , giving ∂ ˆ S ∂ p n = g n = −  4 − h 2 t 2 − h 2 p n  − 1 2 p n  ℓh ∓ p ℓ 2 h 2 + 4 p n p n  , ∂ ˆ S ∂ p n = − g n = 1 2 p n  ℓh ± p ℓ 2 h 2 + 4 p n p n  . (74) (One has to tak e the upp er ch oice of sign in each case t o get the correct con tinuum limit.) Ha ving found an ˆ S for which the relations (74) hold, S = ( ˆ S + p n − p n ) /h pro vides the generating function in (73).  Corollary 5.2 The discr ete P X X X I V e quation asso ciate d with the alt - d P I I e quation (35) c an b e written either as ± p ℓ 2 h 2 + 4 p n p n ± q ℓ 2 h 2 + 4 p n p n = 2 p n (4 − h 2 t ) 2 − h 2 p n , (75) R ational solutions of discr ete T o da and al t - dP I I 19 or with the squar e r o ot signs r emov e d as  ℓ 2 h 2 + 4 p n p n  ℓ 2 h 2 + 4 p n p n  =  2 p 2 n (4 − h 2 t ) 2 (2 − h 2 p n ) 2 − ℓ 2 h 2 − 2 p n ( p n + p n )  2 . Pro of: The second order recurrence relatio n for p n is obtained b y downs hifting the second of (74) and equating it to minus the first.  Remark. T he 3- p oin t corresp ondence (7 5) is equiv alen t to a n analogous equation for the v aria ble y n , that is equation (5.3) in [12], and the equation (75) ha s the same structure as certain discrete Ermak o v-Pinney equations constructed in [33]. The presence o f this structure is due to the connection with discrete Sc hr¨ odinger equations (for whic h, see the pro of of Prop osition 6.1 b elo w). 6. Lax pair for al t - dP I I The con tin uum P I I is equiv alen t to t he pair of bilinear equations (30). A La x pair for P I I is giv en b y the linear problem ∂ Ψ ∂ t = B Ψ , ∂ Ψ ∂ η = A Ψ , (76) where B = − η 2 − iτ n +1 2 τ n iτ n − 1 2 τ n η 2 ! (77) and A =   − η 2 − t 2 + τ n − 1 τ n +1 2 τ 2 n − iητ n +1 τ n ( t ) + i d dt  τ n +1 τ n  iητ n − 1 τ n + i d dt  τ n − 1 τ n  + η 2 + t 2 − τ n − 1 τ n +1 2 τ 2 n   . (78) Consistency of (76) requires ∂ B ∂ η − ∂ A ∂ t + [ B , A ] = 0 , (79) leading to the tw o conditions ¨ φ ± + V φ ± = 0 , (80) where w e ha v e set φ ± = ∓ iτ n ± 1 2 τ n , V = t 2 − 2 φ + φ − . The choice of gauge V = t/ 2 − 2 φ + φ − = 2 d 2 dt 2 log τ n giv es precisely the equation (3 4), and with this normalisation for the tau-functions the conditions (8 0) give the first equation in (30) f or F = 0, together with the same equation for n → n + 1. The bilinear equation (33) is a consequence, and the second equation in (3 0) then follow s. It is w ell known that (7 9) is a n isomono dro my condition: the mono dromy of the solutions o f the second linear equation (76) in the complex η plane is indep enden t of t if a nd only if P I I holds. In the discrete case the situation is completely analogous, based on a linear pro blem that comes from the first part of the discrete T o da La x pair (5). R ational solutions of discr ete T o da and al t - dP I I 20 Prop osition 6.1 The line ar pr oblem Ψ = B Ψ , λ∂ λ Ψ = A Ψ (81) wher e B = 1 λ hτ n +1 2 iτ n − hτ n − 1 2 iτ n λ ! = 1 λ φ + φ − λ ! (82) and A =   2 h 3  − 1 λ 2 + 2( φ + φ − + 1) − λ 2  − t h + 1 2 4 h 3  − φ + λ + φ + λ  4 h 3  − φ − λ + φ − λ  2 h 3  1 λ 2 − 2( φ + φ − + 1) + λ 2  + t h − 1 2   c onstitutes a L a x p air fo r the alt - dP I I e quation (35), which is e quivalent to the c onsistency c ondition λ∂ λ B + B A − AB = 0 (83) for (81). In the limi t h → 0 , this line ar system r e duc es to the L ax p air (76) for P I I . Pro of: The consistency condition (83) implies t he t w o relations φ ± + V φ ± + φ ± = 0 , (84) whic h tak e the form of discrete Sc hr¨ odinger equations with V = ( h 2 t − 4) 2(1 − φ + φ − ) = 2 τ 2 n ( h 2 t − 4) 4 τ 2 n − h 2 τ n +1 τ n − 1 . If w e fix the gauge so that V = − 2 τ 2 n / ( τ n τ n ), then w e g et precisely the relation (49), and up on substituting the lat t er expression for V in to eac h of the equations (84) in turn w e find that (45) and (48) hold. Then b y Theorem 4.1, g n = − τ n − 1 τ n / ( τ n − 1 τ n ) satisfies the al t - dP I I equation. F or the contin uum limit one should tak e λ = e hη 2 , whic h give s A = 2 h A + O ( h 0 ) , B = 1 + h B + O ( h 2 ) , (85) so that the condition ( 79) arises from (83 ) a s h → 0.  Remark. In [1 2] a differen t 2 × 2 Lax pair is presen ted for the al t - dP I I equation, by reduction from the mo dified Boussinesq lattice. Ho we v er, w e ha ve not found a direct relationship b et w een these tw o Lax pairs. 7. Concluding remarks There are man y wa ys to construct a discretisation of a giv en in tegrable differen tial equation, dep ending on whic h par t icular prop erties (e.g. Lax pair, explicit solutions, P oisson structure, Hirota bilinear form, . . . ) one most wishes to preserv e. (F or a thorough accoun t of the Hamiltonian approac h, see [34].) Due to the non- uniqueness of discretisation, it is not alw a ys clear what is the “b est” discrete analogue o f a contin uous system. The deriv ation of the al t - dP I I equation presen ted here w as initially motiv ated b y the construction of exact rational solutions, but it ha s turned o ut t ha t analogues R ational solutions of discr ete T o da and al t - dP I I 21 of all the other structures asso ciated with the second P ainlev ´ e equation a rise naturally here as well. The fact that this discretisation sc heme for P I I , based on the discrete T o da lattice, turned out to b e connected with the sup erp osition form ula fo r P I I I w as completely unexp ected b y us, but led to differen t expressions for the p olynomial tau- functions in terms of Jacobi-T rudi determinan ts. In [12] other solutions of al t - dP I I w ere constructed in terms of Casorati determinants o f discrete Airy functions. In future w e should like to analyse other solutions of the equation (3 5). W e constructed the rational solutions of this equation from p olynomial tau-functions given by Hank el determinan ts, but recen t results for the contin uous case [7] lead us to exp ect that all tau-functions should ha v e a similar structure. Ac kno wledgmen ts W e ar e grateful to P eter Clarkson for inte resting conv ersations on related matters. W e w ould lik e to thank all the r eferees for t heir instructiv e commen ts, in particular for helping us to identify the links b et w een the discrete Y ablonskii-V orob’ev p o lynomials, alt - P I I , P I I I and the Umem ura p olynomials, and for p oin ting out imp ortant references including [12]. App endix Here we pr esen t the pro of o f Theorem 4.4. The result essen tially follo ws from the fact that the B¨ ac klund tra nsformation for P I I I generates a sequence of rationa l solutions w n giv en b y (56) , whic h simultaneously provide r a tional solutions of the al t - dP I I equation (53) b y setting g n = − 1 /w n with z n = n + 1 / 2 for n ∈ Z and µ = − ν . On the other hand, Theorem 4.2 and Corollary 5.1 together imply t ha t the al t - dP I I equation in the form (69) ha s ratio na l solutions, giv en by suitable ratios of discrete Y ablonskii-V orob’ev p olynomials, when ℓ = n + 1 / 2 with n ∈ Z . Up o n comparing (53) with (69) we see that these tw o sets of ratio nal solutions coincide if w e iden tify x = 4 /h 3 , ν = t/h − 4 /h 3 , and then it follows from Theorem 4.3 and Theorem 4 .2 t ha t w n = Z n − 1 ( t, h ) Z n ( t − h, h ) Z n − 1 ( t − h, h ) Z n ( t, h ) = Y n − 1 ( t, h ) Y n ( t − h, h ) Y n − 1 ( t − h, h ) Y n ( t, h ) (A.1) for all n ∈ Z , where Z n ( t, h ) denotes t he right hand side of (57) for n ≥ 0, and the iden tity extends to negative n up on setting Z − n = Z n − 1 . Then it is necessary to sho w that Y n ( t, h ) = Z n ( t, h ) for all n . It is sufficien t to consider n ∈ N as in Theorem 4.4, as the extension to nega t iv e n is trivial. F o r n = 0 the result Y 0 = 1 = Z 0 is o b vious, so to use induction we assume that Y N − 1 = Z N − 1 and then from (A.1) with n = N it holds that Z N ( t − h, h ) Z N ( t, h ) = Y N ( t − h, h ) Y N ( t, h ) . (A.2) Both sides of the relation (A.2 ) are ratios of p olynomials in t (f or Z N this follows from the determinant formula for D N ), and the rational functions o n each side must hav e the R ational solutions of discr ete T o da and al t - dP I I 22 same zero es a nd p oles. Ho w ev er, a lthough Theorem 3.1 implies that the numerator and denominator on the rig h t hand side hav e no common factors, w e cannot immediately assert that the same is true on the left hand side without kno wing the degree of D N ( x, ν ) in ν . (The pro of of Corollary 2 in [1 3] just give s the degree in x of D N , denoted F N there, as N ( N + 1) / 2.) Ho w ev er, by Pro p osition 3 in [13] these Jacobi-T rudi determinants satisfy the recurrence (2 n + 1) D n +1 D n − 1 + x ( D n D ′′ n − ( D ′ n ) 2 ) + D n D ′ n − ( x + ν ) D 2 n = 0 , (A.3) with D − 1 = 1 = D 0 and ′ denoting d/ d x . 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