Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations
aa r X i v : . [ m a t h - ph ] M a y DEGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATORAND q -PAINLEV ´E EQUATIONS KOUICHI TAKEMURA
Abstract.
It is known that the Painlev´e VI is obtained by connection preservingdeformation of some linear differential equations, and the Heun equation is obtainedby a specialization of the linear differential equations. We inverstigate degenerationsof the Ruijsenaars-van Diejen difference opearators and show difference analoguesof the Painlev´e-Heun correspondence. Introduction
In this paper, we investigate q -difference equations that are generalisations of theHeun equation and the Painlev´e VI equation.Heun’s differential equation is given by d ydz + (cid:18) γz + δz − ǫz − t (cid:19) dydz + αβz − qz ( z − z − t ) y = 0 , (1.1)with the condition γ + δ + ǫ = α + β +1, and it is a standard form of Fuchsian differentialequation with four singularities { , , t, ∞} . Note that the Gauss hypergeometricequation is a standard form of Fuchsian differential equation with three singularities { , , ∞} . The Heun equation has an accessory parameter q which is independentfrom local exponents, although the hypergeometric equation does not have it.It is known that the Heun equation admits an expression in terms of elliptic func-tions. Let ℘ ( x ) be the Weierstrass elliptic function with basic periods (2 ω , ω ). Put ω = − ω − ω , ω = 0 and e i = ℘ ( ω i ) ( i = 1 , , z = ℘ ( x ) − e e − e , t = e − e e − e and applying a gauge transformation, we obtain an elliptical representation of Heun’sdifferential equation (see [17]): − d dx + X i =0 l i ( l i + 1) ℘ ( x + ω i ) ! f ( x ) = Ef ( x ) . (1.3)Here the coupling constants l , . . . , l correspond to the parameters α, . . . , ǫ in Eq.(1.1)and the eigenvalue E corresponds to the accessory parameter q . Mathematics Subject Classification.
Key words and phrases.
Ruijsenaars system, degeneration, Painlev´e equation, Heun equation.
The Painlev´e VI equation is a non-linear ordinary differential equation given by d λdt = 12 (cid:18) λ + 1 λ − λ − t (cid:19) (cid:18) dλdt (cid:19) − (cid:18) t + 1 t − λ − t (cid:19) dλdt (1.4) + λ ( λ − λ − t ) t ( t − (cid:26) α + β tλ + γ ( t − λ − + δ t ( t − λ − t ) (cid:27) . See [2] for a review of the Painlev´e equations. In particular, it is known that solutionsof the Painlev´e VI equation do not have movable singularities other than poles, thatis called the Painlev´e property. Painlev´e VI is also obtained by monodromy pre-serving deformation of the 2 × { , , t, ∞} . The Fuchsian system of equations is equivalent to the following Fuchsianequation d y dz + (cid:18) − θ z + 1 − θ z − − θ t z − t − z − λ (cid:19) dy dz (1.5) + (cid:18) κ ( κ + 1) z ( z −
1) + λ ( λ − µz ( z − z − λ ) − t ( t − Hz ( z − z − t ) (cid:19) y = 0 ,H = 1 t ( t −
1) [ λ ( λ − λ − t ) µ − { θ ( λ − λ − t )+ θ λ ( λ − t ) + ( θ t − λ ( λ − } µ + κ ( κ + 1)( λ − t )] . Note that the singularity z = λ is apparent, which follows from the equality for H .The monodromy of the solution to Eq.(1.5) is preserved as the parameter t varies,if there exist rational functions a ( z, t ) and a ( z, t ) of the variable z such that theequation(1.6) ∂y∂t = a ( z, t ) y + a ( z, t ) ∂y∂z is compatible to Eq.(1.5) (see [2]). It follows from a lengthy calculation that thecompatibility condition is equivalent to dλdt = ∂H∂µ , dµdt = − ∂H∂λ , (1.7)which is called the Painlev´e VI system. By eliminating µ , we obtain the Painlev´e VIequation.Recall that Eq.(1.5) has five singularities { , , t, ∞ , λ } , and the singularity z = λ is superfluous for the Heun equation. By specializing the point z = λ to regularsingularities { , , t, ∞} , we may derive the Heun equation. For example, by setting λ = t in Eq.(1.5) we have d y dz + (cid:18) − θ z + 1 − θ z − − θ t z − t (cid:19) dy dz + κ ( κ + 1)( z − t ) + θ t t ( t − µz ( z − z − t ) y = 0 . (1.8)Therefore the Heun equation is related with the Painlev´e VI equation through thelinear differential equation given by Eq.(1.5). We can also obtain the Heun equationby other specializations, and they are related with the space of initial conditions EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 3 (see [16]). See also [14, 15] for other perspectives on relationship between the Heunequation and the Painlev´e VI equation. Note that the Painlev´e VI equation alsoadmits elliptical representations [5, 1, 15, 21], which were applied in various ways.In this paper, we propose a difference analogue of the correspondence between theHeun equation and the Painlev´e VI equation.Sakai [12] investigated difference analogue of the Painlev´e equation by using struc-tures of some algebraic surfaces which are generalisations of the space of initial con-ditions, and proposed a list of the equations. There are three kinds of differencePainlev´e equations, i.e. elliptic difference, q -difference (or multiplicative difference)and additive difference, and each difference equation is labelled by some affine rootsystems from its symmetry. The q -difference Painlev´e equations of types E (1)7 , E (1)6 and D (1)5 are at issue in this paper.Before giving a difference analogue of the Heun equation, we discuss a multivariablegeneralization of the Heun equation. The quantum Inozemtsev system of type BC N is a quantum mechanical N -particle system whose Hamiltonian is given by H = − N X j =1 ∂ ∂x j + 2 l ( l + 1) X ≤ j 0, weobtain the Hamiltonian of the Inozemtsev system [18, 9]. It is known that commutingoperators of the Ruijsenaars-van Diejen system exist as is the case of the Inozemtsevsystem [4]. We may regard the Ruijsenaars-van Diejen operator with one variable as a KOUICHI TAKEMURA difference analogue of the Heun equation. It is known that the Ruijsenaars-van Diejenoperator has E spectral symmetry [11]. On the other hand, the elliptic differencePainlev´e equation admits E (1)8 symmetry [12]. We expect to clarify relationshipsbetween the one variable difference equation of Ruijsenaars-van Diejen type and theelliptic difference Painlev´e equation.In this paper we investigate degenerations of the Ruijsenaars-van Diejen opera-tor and find correspondences with linear q -difference equations which are relatedwith q -difference Painlev´e equations. We find that we can take degenerations of theRuijsenaars-van Diejen operator of N variables four times, although it seems thatthe first two were essentially obtained by van Diejen [18]. The degenerations arestill interesting in the setting of one variable. By taking degeneration four times, weobtain the following q -difference operator A h i ( x ): A h i ( x ) g ( x ) = x − ( x − h q / )( x − h q / ) g ( x/q ) + x − l l ( x − l q − / )( x − l q − / ) g ( qx )(1.12) − { ( l + l ) x + ( l l l l h h ) / ( h / + h − / ) x − } g ( x ) . Then we may regard the equation(1.13) A h i ( x ) g ( x ) = Eg ( x ) ( E : eigenvalue)as a q -deformation of Heun equation (1.1). On the other hand, Eq.(1.13) is obtainedas a special case of the linear q -difference equation by Jimbo and Sakai [3] which isrelated with the q -Painlev´e VI equation by the connection preserving deformation.Similarly the equations for eigenfunctions of the second degenerate operator and thethird degenerate operator are also obtained as special cases of the linear q -differenceequations obtained by Yamada [20] which are related with the q -Painlev´e equationsof type E (1)6 and type E (1)7 .This article is organized as follows. In section 2, we apply degeneration of theRuijsenaars-van Diejen operator with one variable four times. In section 3, we reviewlinear q -difference equations which are related with some q -Painlev´e equations andobtain the degenerated Ruijsenaars-van Diejen operators with one variable by special-izing the parameters. In section 4, we extend the degeneration to the multivariablecase. In section 5, we propose some problems related with results in this paper.2. Degeneration of Ruijsenaars-van Diejen operator with onevariable Ruijsenaars-van Diejen operator. We describe the Ruijsenaars-van Diejenoperator with one variable explicitly. Let a + , a − be complex numbers whose realparts are positive and R ± ( z ) be the functions defined by(2.1) R ± ( z ) = ∞ Y k =1 (1 − q k − ± e πiz )(1 − q k − ± e − πiz ) , q ± = e − πa ± . They are modified versions of theta functions with the half periods 1 / ia ± / A + ( h ; z ) = V + ( h ; z ) exp( − ia − ∂ z ) + V + ( h ; − z ) exp( ia − ∂ z ) + U + ( h ; z ) , EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 5 where V + ( h ; z ) = Q n =1 R + ( z − h n − ia − / R + (2 z + ia + / R + (2 z − ia − + ia + / , (2.3) U + ( h ; z ) = P t =0 p t, + ( h )[ E t, + ( µ ; z ) − E t, + ( µ ; ω t, + )]2 R + ( µ − ia + / R + ( µ − ia − − ia + / , and we are using ω , + = 0 , ω , + = 1 / , ω , + = ia + / , ω , + = − / − ia + / , (2.4) p , + ( h ) = Y n =1 R + ( h n ) , p , + ( h ) = e − πa + Y n =1 e − iπh n R + ( h n − ia + / ,p , + ( h ) = Y n =1 R + ( h n − / , p , + ( h ) = e − πa + Y n =1 e iπh n R + ( h n + 1 / ia + / , E t, + ( µ ; z ) = R + ( z + µ − ia + / − ia − / − ω t, + ) R + ( z − µ + ia + / ia − / − ω t, + ) R + ( z − ia + / − ia − / − ω t, + ) R + ( z + ia + / ia − / − ω t, + ) , ( t = 0 , , , . We adapt the expression in [11], which is slightly different from theone in [10] with an additive constant. Note that the function U + ( h ; z ) is independentfrom the parameter µ in the case of one variable z , which can be proved as the firstpart of Lemma 3.2 in [9]. Hence the operator A + ( h ; z ) is also independent from theparameter µ .We can obtain an elliptical representation of the Heun equation (1.3) from theequation A + ( h ; z ) f ( z ) = Ef ( z ) ( E : eigenvalue) by taking a suitable limit as a − → First degeneration. We are going to take a trigonometric limit ( q + → 0) ofthe Ruijsenaars-van Diejen operator with one variable. The function R + ( z ) satisfies(2.5) R + ( z ∓ ia + ) = − e πa + e ± πiz R + ( z )and we have the following expansion as q + → a + → + ∞ ): R + ( z ) = 1 − ( e πiz + e − πiz ) q + + q + O ( q ) , (2.6) R + ( z ± ia + / 2) = (1 − e ∓ πiz )(1 − ( e πiz + e − πiz ) q + O ( q )) . We set h n = ˜ h n − ia + / 2. Then the function V + ( h ; z ) admits the following limit as q + → V + ( h ; z ) → V h i ( h ; z ) = Q n =1 (1 − e − πiz e πi ˜ h n e − πa − )(1 − e − πiz )(1 − e − πiz e − πa − ) . (2.7)By considering the limit of the function U + ( h ; z ) as q + → 0, we have the followingproposition. KOUICHI TAKEMURA Proposition 2.1. Let A ( h, q + ; z ) be the Ruijsenaars-van Diejen operator defined inEq.(2.2). As q + → , we have (2.8) A ( h, q + ; z ) + Q n =1 e πi ˜ h n (1 − e πa − ) q − + C ! f ( z ) → A h i ( h ; z ) f ( z ) for any f ( z ) , where (2.9) A h i ( h ; z ) = V h i ( h ; z ) exp( − ia − ∂ z ) + V h i ( h ; − z ) exp( ia − ∂ z )) + U h i ( h ; z ) ,V h i ( h ; z ) was defined in Eq.(2.7), U h i ( h ; z ) = Q n =1 ( e πi ˜ h n − − e πiz e πa − )(1 − e − πiz e πa − ) + Q n =1 ( e πi ˜ h n + 1)2(1 + e πiz e πa − )(1 + e − πiz e πa − )(2.10)+ e − πa − Y n =1 e πi ˜ h n · h ( e πiz + e − πiz ) X n =1 ( e πi ˜ h n + e − πi ˜ h n ) − ( e πa − + e − πa − )( e πiz + e − πiz ) i , and C = Y n =1 e πi ˜ h n · h e − πa − ( e πa − + e − πa − )(2.11)+ 12 + X ≤ n 1) + Y n =1 ( e πi ˜ h n + 1) o (1 − e πa − ) i . Proof. It follows from R + ( z ± ia + / 2) = 1 − e ∓ πiz + O ( q ) that p , + ( h ) E , + ( µ ; z )(2.12)= R + ( z + µ − ia + / − ia − / R + ( z − µ + ia + / ia − / R + ( z − ia + / − ia − / R + ( z + ia + / ia − / Y n =1 R + (˜ h n − ia + / − e πi ( z + µ − ia − / )(1 − e − πi ( z − µ + ia − / )(1 − e πi ( z − ia − / )(1 − e − πi ( z + ia − / ) Y n =1 (1 − e πi ˜ h n ) + O ( q )= n e πiµ + ( e πiµ − e πiµ e πa − − − e πiz e πa − )(1 − e − πiz e πa − ) o Y n =1 ( e πi ˜ h n − 1) + O ( q ) , EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 7 p , + ( h ) E , + ( µ ; z )(2.13) = n e πiµ + ( e πiµ − e πiµ e πa − − e πiz e πa − )(1 + e − πiz e πa − ) o Y n =1 ( e πi ˜ h n + 1) + O ( q ) ,R + ( µ − ia + / R + ( µ − ia − − ia + / 2) = (1 − e πiµ )(1 − e πiµ e πa − ) + O ( q ) . Hence p , + ( h )( E , + ( µ ; z ) − E , + ( µ ; ω , + ))2 R + ( µ − ia + / R + ( µ − ia − − ia + / 2) + p , + ( h )( E , + ( µ ; z ) − E , + ( µ ; ω , + ))2 R + ( µ − ia + / R + ( µ − ia − − ia + / n − e πiz e πa − )(1 − e − πiz e πa − ) − − e πa − ) o Y n =1 ( e πi ˜ h n − n e πiz e πa − )(1 + e − πiz e πa − ) − − e πa − ) o Y n =1 ( e πi ˜ h n + 1) + O ( q ) . Set ˜ p ( h ) = 8 + X ≤ n 3, we have E t, + ( µ ; ω t, + ) = R + ( − µ + ia − / ia + / R + ( ia − / ia + / (2.20) = (1 − e πiµ e πa − ) (1 − e πa − ) [1 + 2( e πa − − e − πa − e − πiµ )(1 − e πiµ ) q + O ( q )] . Then p , + ( h ) E , + ( µ ; ω , + ) + p , + ( h ) E , + ( µ ; ω , + )(2.21)= 2 (1 − e πiµ e πa − ) (1 − e πa − ) Y n =1 e πi ˜ h n · [ q − + ˜ p ( h ) + 2( e πa − − e − πa − e − πiµ )(1 − e πiµ ) + O ( q + )] . EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 9 By combining with R + ( µ − ia + / R + ( µ − ia − − ia + / − e πiµ )(1 − e πiµ e πa − )(1 − ( e πa − e πiµ + e − πa − e − πiµ + e πiµ + e − πiµ ) q + O ( q )) , we have p , + ( h )( E , + ( µ ; z ) − E , + ( µ ; ω , + )) + p , + ( h )( E , + ( µ ; z ) − E , + ( µ ; ω , + ))2 R + ( µ − ia + / R + ( µ − ia − − ia + / Y n =1 e πi ˜ h n · h e − πa − n ( e πiz + e − πiz ) X n =1 ( e πi ˜ h n + e − πi ˜ h n ) − ( e − πa − + e πa − )( e πiz + e − πiz ) o − q − (1 − e πa − ) − ˜ p ( h ) + 4(1 − e πa − ) − e − πa − ( e − πa − + e πa − ) + O ( q + ) i . Therefore P k =0 p k, + ( h )( E k, + ( µ ; z ) − E k, + ( µ ; ω k, + ))2 R + ( µ − ia + / R + ( µ − ia − − ia + / Q n =1 e iπ ˜ h n ( e πi ˜ h n − e − πi ˜ h n )2(1 − e πiz e πa − )(1 − e − πiz e πa − ) + Q n =1 e iπ ˜ h n ( e πi ˜ h n + e − πi ˜ h n )2(1 + e πiz e πa − )(1 + e − πiz e πa − )+ Y n =1 e πi ˜ h n h e − πa − ( e πiz + e − πiz ) X n =1 ( e πi ˜ h n + e − πi ˜ h n ) − e − πa − ( e − πa − + e πa − )( e πiz + e − πiz ) − q − (1 − e πa − ) − e − πa − ( e − πa − + e πa − ) − 12 + X ≤ n 1) + Y n =1 ( e πi ˜ h n + 1) o (1 − e πa − ) i . (cid:3) Note that the operator A h i ( h ; z ) does not contain the parameter µ , and it is notsurprising because the Ruijsenaars-van Diejen operator A ( h, q + ; z ) is independentfrom the parameter µ despite it appears in the expression.To obtain the second degeneration in section 2.3, we apply a gauge transformationto the operator in Proposition 2.1 by using the function R − ( z ) defined in Eq.(2.1),which satisfies(2.25) R − ( z ∓ ia − ) = − e πa − e ± πiz R − ( z ) . By the gauge transformation(2.26) ˜ A h i ( h, z ) = R − ( z ) − ◦ A h i ( h, z ) ◦ R − ( z ) , we have the following operator:(2.27) ˜ A h i ( h ; z ) = ˜ V h i ( h ; z ) exp( − ia − ∂ z ) + ˜ W h i ( h ; z ) exp( ia − ∂ z ) + U h i ( h ; z ) , where U h i ( h ; z ) was defined in Eq.(2.10) and˜ V h i ( h ; z ) = Q n =1 (1 − e − πiz e πi ˜ h n e − πa − ) e − πa − e − πiz (1 − e − πiz )(1 − e − πiz e − πa − ) , (2.28) ˜ W h i ( h ; z ) = Q n =1 (1 − e πiz e πi ˜ h n e − πa − ) e − πa − e πiz (1 − e πiz )(1 − e πiz e − πa − ) . Note that this operator was essentially obtained by van Diejen [18] in the multivariablecase.2.3. Second degeneration. We investigate a degeneration of the operator given byEq.(2.27). Proposition 2.2. In Eq.(2.27), we replace z by z + iR , ˜ h n ( n = 1 , , , by h n + iR , ˜ h n ( n = 5 , , , by h n − iR and take the limit R → + ∞ . Then we arrive at theoperator (2.29) A h i ( h ; z ) = V h i ( h ; z ) exp( − ia − ∂ z ) + W h i ( h ; z ) exp( ia − ∂ z ) + U h i ( h ; z ) , where V h i ( h ; z ) = e πiz Y n =1 (1 − e − πiz e πih n e − πa − ) Y n =5 e πih n , (2.30) W h i ( h ; z ) = e πa − e − πiz Y n =5 (1 − e πiz e πih n e − πa − ) ,U h i ( h ; z ) = Y n =5 e πih n h(cid:16) X n =1 e πih n + X n =5 e − πih n (cid:17) e − πa − e πiz − (1 + e − πa − ) e πiz i + Y n =1 e πih n h(cid:16) X n =1 e − πih n + X n =5 e πih n (cid:17) e − πa − e − πiz − (1 + e − πa − ) e − πiz i . Namely we have (2.31) e − πR ˜ A h i ( h + iRv ; z + iR ) f ( z ) → A h i ( h ; z ) f ( z ) as R → + ∞ for any f ( z ) , where v = (1 , , , , − , − , − , − .Proof. We define the equivalence a ∼ b by lim R → + ∞ a/b = 1. Then˜ V h i ( h + iRv ; z + iR )(2.32) = Q n =1 (1 − e − πiz e πih n e − πa − ) Q n =5 (1 − e − πiz e πR e πih n e − πa − ) e − πa − e − πiz e πR (1 − e − πiz e πR )(1 − e − πiz e πR e − πa − ) ∼ e πiz e πR Y n =1 (1 − e − πiz e πih n e − πa − ) Y n =5 e πih n , EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 11 ˜ W h i ( h + iRv ; z + iR )(2.33) = Q n =1 (1 − e πiz e πih n e − πR e − πa − ) Q n =5 (1 − e πiz e πih n e − πa − ) e − πa − e πiz e − πR (1 − e πiz e − πR )(1 − e πiz e − πR e − πa − ) ∼ e πa − e − πiz e πR Y n =5 (1 − e πiz e πih n e − πa − ) , ˜ U h i ( h + iRv ; z + iR )(2.34)= − e πR e − πa − e πiz Y n =5 e πih n Q n =1 (1 − e πih n e − πR ) Q n =5 (1 − e − πR e − πih n )2(1 − e πiz e − πR e πa − )(1 − e − πa − e πiz e − πR )+ e πR e − πa − e πiz Y n =5 e πih n Q n =1 (1 + e πih n e − πR ) Q n =5 (1 + e − πR e − πih n )2(1 + e πiz e − πR e πa − )(1 + e − πa − e πiz e − πR )+ e − πa − Y n =1 e πih n hn X n =1 ( e πih n e − πR + e − πih n e πR ) + X n =5 ( e πih n e πR + e − πih n e − πR ) o · ( e πiz e − πR + e − πiz e πR ) − ( e − πa − + e πa − )( e πiz e − πR + e − πiz e πR ) i ∼ · e πR + e πR Y n =5 e πih n n(cid:16) X n =1 e πih n + X n =5 e − πih n (cid:17) e − πa − e πiz − (1 + e − πa − ) e πiz o + e πR e − πa − Y n =1 e πih n n(cid:16) X n =1 e − πih n + X n =5 e πih n (cid:17) e − πiz − ( e − πa − + e πa − ) e − πiz o . Thus the proposition is obtained. (cid:3) Set l n = − h n +4 ( n = 1 , , , A h i ( h, l ; z ) = e πa − Y n =5 e − πih n · e πiz ◦ A h i ( h, z ) ◦ e − πiz , we have the following operator:˜ A h i ( h, l ; z ) = ˜ V h i ( h ; z ) exp( − ia − ∂ z ) + ˜ W h i ( l ; z ) exp( ia − ∂ z ) + ˜ U h i ( h, l ; z ) , (2.36) where˜ V h i ( h ; z ) = e πiz Y n =1 (1 − e πih n e − πa − e − πiz ) , ˜ W h i ( l ; z ) = e πiz Y n =1 (1 − e πil n e πa − e − πiz ) , (2.37)˜ U h i ( h, l ; z ) = X n =1 ( e πih n + e πil n ) e πiz − ( e πa − + e − πa − ) e πiz + Y n =1 e πi ( h n + l n ) h X n =1 ( e − πih n + e − πil n ) e − πiz − ( e πa − + e − πa − ) e − πiz i . Note that this operator was also essentially obtained in [18].Set x = e πiz , q = e − πa − , and replace e πih n and e πil n by h n and l n . Then thedifference operator ˜ A h i ( h, l ; z ) is written as A h i ( x ) g ( x ) = x − Y n =1 ( x − h n q / ) g ( x/q ) + x − Y n =1 ( x − l n q − / ) g ( qx ) + U ( x ) g ( x ) , (2.38) U ( x ) = − ( q / + q − / ) x + X n =1 ( h n + l n ) x + Y n =1 h / n l / n · [ − ( q / + q − / ) x − + X n =1 ( h − n + l − n ) x − ] . The equation A h i ( x ) g ( x ) = Eg ( x ) is also obtained as a specialization of the lineardifference equation which is related with the q -Painlev´e equation of type E (1)7 in [20].We discuss it in section 3.3.2.4. Third degeneration.Proposition 2.3. In Eq.(2.36), we replace z by z − iR , h n ( n = 1 , by h n − iR , h n ( n = 3 , by h n + iR , l n ( n = 1 , , , by l n − iR and take the limit R → + ∞ .Then we arrive at the operator A h i ( h, l ; z ) = V h i ( h ; z ) exp( − ia − ∂ z ) + W h i ( l ; z ) exp( ia − ∂ z ) + U h i ( h, l ; z ) , (2.39) EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 13 where V h i ( h ; z ) = e πiz Y n =1 (1 − e πih n e − πa − e − πiz ) , (2.40) W h i ( l ; z ) = e πiz Y n =1 (1 − e πil n e πa − e − πiz ) ,U h i ( h, l ; z ) = (cid:16) X n =1 e πih n + X n =1 e πil n (cid:17) e πiz − ( e πa − + e − πa − ) e πiz + e πih e πih ( e πi ( h − h ) + e πi ( h − h ) ) Y n =1 e πil n · e − πiz . Namely, as R → + ∞ we have (2.41) e − πR ˜ A h i ( h − iRv , l − iRv ; z − iR ) f ( z ) → A h i ( h, l ; z ) f ( z ) for any f ( z ) , where v = (1 , , − , − , v = (1 , , , .Proof. We have˜ V h i ( h − iRv ; z − iR )(2.42) = e πiz e πR Y n =1 (1 − e − πiz e πih n e − πa − ) Y n =3 (1 − e − πR e − πiz e πih n e − πa − ) ∼ e πR e πiz Y n =1 (1 − e − πiz e πih n e − πa − ) , ˜ W h i ( l − iRv ; z − iR ) = e πiz e πR Y n =1 (1 − e − πiz e πil n e πa − ) , (2.43)˜ U h i ( h − iRv , l − iRv ; z − iR ) = − (cid:16) X n =1 e πih n e πR + X n =3 e πih n e − πR (2.44)+ X n =1 e πil n e πR (cid:17) e πiz e πR + ( e πa − + e − πa − ) e πiz e πR + e πR Y n =1 e πih n Y n =5 e πil n n(cid:16) X n =1 e − πih n e − πR + X n =3 e − πih n e πR + X n =1 e − πil n e − πR (cid:17) e − πiz e − πR − ( e πa − + e − πa − ) e − πiz e − πR o ∼ e πR n − (cid:16) X n =1 e πih n + X n =1 e πil n (cid:17) e πiz + ( e πa − + e − πa − ) e πiz + (cid:16) Y n =1 e πih n e πil n · X n =3 e − πih n (cid:17) e − πiz o . Thus the proposition is obtained. (cid:3) Set x = e πiz , q = e − πa − , and replace e πih n and e πil n by h n and l n . Then thedifference operator A h i ( h, l ; z ) is written as A h i ( x ) g ( x ) = Y n =1 ( x − h n q / ) g ( x/q ) + x − Y n =1 ( x − l n q − / ) g ( qx ) + U ( x ) g ( x ) , (2.45) U ( x ) = (cid:16) X n =1 h n + X n =1 l n (cid:17) x − ( q / + q − / ) x + ( l l l l h h ) / ( h / h − / + h − / h / ) x − . Let E be a constant. The equation A h i ( x ) g ( x ) = Eg ( x ) is written as Y n =1 ( x − h n q / ) g ( x/q ) + x − Y n =1 ( x − l n q − / ) g ( qx ) + {− ( q / + q − / ) x (2.46)+ (cid:16) X n =1 h n + X n =1 l n (cid:17) x − E + ( l l l l h h ) / ( h / h − / + h − / h / ) x − } g ( x ) = 0 . This equation is also obtained as a specialization of the linear difference equationwhich is related with the q -Painlev´e equation of type E (1)6 in [20]. We discuss it insection 3.2. Set ¯ A h i ( x ) = v ( x ) ◦ A h i ( h, l ; z ) ◦ v ( x ) − , (2.47) v ( x ) = ( x − l q / ; q ) ∞ = ∞ Y k =0 (1 − x − l q / q k ) . We replace h , h and l by ˜ h , 1 and h in Eq.(2.46). Then the gauge transformedequation ¯ A h i ( x ) g ( x ) = Eg ( x ) is written as Y n =1 ( x − h n q / ) g ( x/q ) + Y n =1 ( x − l n q − / ) g ( qx ) + n − ( q / + q − / ) x (2.48) + X n =1 ( h n + l n ) x − Ex + ( l l l h h h ) / (˜ h / + ˜ h − / ) o g ( x ) = 0 . To obtain the fourth degeneration, we apply the gauge transformation(2.49) ˜ A h i ( h, l ; z ) = R − ( z ) ◦ A h i ( h, l ; z ) ◦ R − ( z ) − . Then we have(2.50) ˜ A h i ( h, l ; z ) = ˜ V h i ( h ; z ) exp( − ia − ∂ z ) + ˜ W h i ( l ; z ) exp( ia − ∂ z ) + U h i ( h, l ; z ) , EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 15 where U h i ( h, l ; z ) was defined in Eq.(2.40) and˜ V h i ( h ; z ) = e − πa − Y n =1 (1 − e πih n e − πa − e − πiz ) , (2.51) ˜ W h i ( l ; z ) = e − πa − e πiz Y n =1 (1 − e πil n e πa − e − πiz ) . Fourth degeneration and q -Heun equation.Proposition 2.4. In Eq.(2.50), we replace z by z + iR , h n ( n = 1 , , , by h n + iR , l n ( n = 1 , by l n + iR , l n ( n = 3 , by l n − iR and take the limit R → + ∞ . Thenwe arrive at the operator A h i ( h, l ; z ) = V h i ( h ; z ) exp( − ia − ∂ z ) + W h i ( l ; z ) exp( ia − ∂ z ) + U h i ( h, l ; z ) , (2.52) where V h i ( h ; z ) = e − πa − Y n =1 (1 − e πih n e − πa − e − πiz ) , (2.53) W h i ( l ; z ) = e πiz e πi ( l + l ) 2 Y n =1 (1 − e πil n e πa − e − πiz ) ,U h i ( h, l ; z ) = ( e πil + e πil ) e πiz + e πih e πih ( e πi ( h − h ) + e πi ( h − h ) ) Y n =1 e πil n · e − πiz . Namely, as R → + ∞ we have (2.54) e − πR ˜ A h i ( h + iRv , l + iRv ; z + iR ) f ( z ) → A h i ( h, l ; z ) f ( z ) for any f ( z ) , where v = (1 , , − , − , v = (1 , , , .Proof. We have˜ V h i ( h + iRv ; z + iR ) = e − πa − Y n =1 (1 − e πih n e − πa − e − πiz ) , (2.55) ˜ W h i ( l + iRv ; z + iR ) = e − πR e − πa − e πiz Y n =1 (1 − e πil n e πa − e − πiz ) · (2.56) · Y n =3 (1 − e πil n e πa − e − πiz e πR ) ∼ e πiz Y n =1 (1 − e πil n e πa − e − πiz ) Y n =3 e πil n , ˜ U h i ( h + iRv , l + iRv ; z + iR )(2.57)= (cid:16) X n =1 e πih n e − πR + X n =1 e πil n e − πR + X n =3 e πil n e πR (cid:17) e πiz e − πR − ( e πa − + e − πa − ) e πiz e − πR + Y n =1 e πil n · e πih e πih e − πR ( e πi ( h − h ) + e πi ( h − h ) ) e − πiz e πR ∼ ( e πil + e πil ) e πiz + Y n =1 e πil n · e πih e πih ( e πi ( h − h ) + e πi ( h − h ) ) e − πiz . (cid:3) We apply a gauge transformation to the operator in Proposition 2.1 by using thefunction R − ( z ) defined in Eq.(2.1), which satisfies R − ( z ∓ ia − ) = − e πa − e ± πiz R − ( z ).By the multiplication and the gauge transformation given by(2.58) ˜ A h i ( h, l ; z ) = − R − ( z ) − e − πiz ◦ A h i ( h, l ; z ) ◦ R − ( z ) e πiz , we have ˜ A h i ( h, l ; z ) = ˜ V h i ( h ; z ) exp( − ia − ∂ z ) + ˜ W h i j ( l ; z ) exp( ia − ∂ z ) − U h i ( h, l ; z ) , (2.59)where was defined in Eq.(2.53) and˜ V h i ( h ; z ) = e πiz Y n =1 (1 − e πih n e − πa − e − πiz ) , (2.60) ˜ W h i ( l ; z ) = e πil e πil e πiz Y n =1 (1 − e πil n e πa − e − πiz ) . Set x = e πiz , q = e − πa − , replace e πih n and e πil n by h n and l n , and set h = 1. Thenthe difference operator is written as A h i ( x ) g ( x ) = x − ( x − h q / )( x − h q / ) g ( x/q ) + x − l l ( x − l q − / )( x − l q − / ) g ( qx )(2.61) − { ( l + l ) x + ( l l l l h h ) / ( h / + h − / ) x − } g ( x ) . Let E be a constant. The equation A h i ( x ) g ( x ) = Eg ( x ) is written as( x − h q / )( x − h q / ) g ( x/q ) + l l ( x − l q − / )( x − l q − / ) g ( xq )(2.62) − { ( l + l ) x + Ex + ( l l l l h h ) / ( h / + h − / ) } g ( x ) = 0 . This equation is also obtained as a specialization of the linear difference equationwhich is related with the q -Painlev´e VI equation [3]. We discuss it in section 3.1. EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 17 We call Eq.(2.62) the q -Heun equation, because it has a limit to the Heun equation,which we show in the rest of this subsection. We rewrite Eq.(2.62) as( x − t q h q / )( x − t q h q / ) g ( x/q ) + q l + l ( x − t q l q − / )( x − t q l q − / ) g ( xq )](2.63) − { ( q l + q l ) x − (2( t + t ) + ( q − E + ( q − ˜ E ) x + t t q ( l + l + l + l + h + h ) / ( q h / + q − h / ) } g ( x ) = 0 , where(2.64) E = ( l + l )( t + t ) + ( l + h ) t + ( l + h ) t . Set q = 1 + ε . We divide Eq.(2.63) by ε . By using Taylor’s expansion g ( x/q ) = g ( x ) + ( − ε + ε ) xg ′ ( x ) + ε x g ′′ ( x ) / O ( ε ) , (2.65) g ( qx ) = g ( x ) + εxg ′ ( x ) + ε x g ′′ ( x ) / O ( ε ) , we find the following limit as ε → x ( x − t )( x − t ) g ′′ ( x )(2.66)+ [(1 + h − l ) x ( x − t ) + (1 + h − l ) x ( x − t ) + (˜ l − x − t )( x − t )] xg ′ ( x )+ [( l l e ) x + ˜ Bx + t t (˜ l/ − h / l/ − − h / g ( x ) = 0 , where˜ l = l + l + l + l − h − h , (2.67)˜ B = ˜ E − t n h + ( l + l + l − − o + t n h + ( l + l + l − − o . This equation is a Fuchsian differential equation with four singularities { , t , t , ∞} and the local exponents are given by the following Riemann scheme:(2.68) x = 0 x = t x = t x = ∞ − ˜ l/ h / l − ˜ l/ − h / l − h l − h l By setting z = x/t and g ( x ) = x − ˜ l/ − h / ˜ g ( z ), the function y = ˜ g ( z ) satisfies Heun’sdifferential equation; d ydz + (cid:18) γz + δz − ǫz − t (cid:19) dydz + αβz − qz ( z − z − t ) y = 0 , (2.69)where t = t /t , γ = 1 − h , δ = 1 + h − l , ǫ = 1 + h − l , { α, β } = { l + 1 − ˜ l/ − h / , l + 1 − ˜ l/ − h / } and t = t /t . Therefore it is reasonable to call Eq.(2.62)the q -Heun equation. Linear q -difference equations related with q -Painlev´e equations It is widely known that the Lax formalism is a powerful method for studyingsoliton equations, and a similar method is applied for Painlev´e-type equations. Inparticular Jimbo and Sakai [3] obtained q -Painlev´e VI (or the q -Painlev´e equationof type D (1)5 ) by finding ”Lax forms”. We may regard the Lax forms for differencePainlev´e equations as a linear difference equation and an associated deformationequation where the difference Painlev´e equation is written in the form of compatibilityof them (see [13]).About ten years later from the discovery of q -Painlev´e VI, Sakai [13] presenteda problem to find Lax forms for difference Painlev´e equations in his list [12], andYamada made a significant contribution for the problem. Namely, Yamada discoveredLax forms for the q -difference Painlev´e equations of types D (1)5 , E (1)6 , E (1)7 , E (1)8 [20]and the elliptic difference Painlev´e equation [19]. For type D (1)5 , it essentially coincideswith the one by Jimbo and Sakai [3].In this section, we observe that our degenerete operators of Ruijsenaars-van Diejenappear by restricting the parameters in the linear q -differential equations related to q -Painlev´e equations of types D (1)5 , E (1)6 , E (1)7 .3.1. Linear q -difference equation related with q -Painlev´e VI. Jimbo and Sakai[3] found a Lax form for the q -Painlev´e VI by considering connection preservingdeformation of the linear system of q -differential equations Y ( qx, t ) = A ( x, t ) Y ( x, t ) . (3.1)To derive q -Painlev´e VI, they rewrite the condition for connection preserving de-formation as compatibility of Eq.(3.1) with a deformation equation of the form Y ( x, qt ) = B ( x, t ) Y ( x, t ), which is a Lax form of q -Painlev´e VI.We focus on Eq.(3.1). The 2 × A ( x, t ) is taken in the form Eqs.(9)-(11)in [3], i.e. A ( x, t ) = A ( t ) + A ( t ) x + A x = (cid:18) a ( x ) a ( x ) a ( x ) a ( x ) (cid:19) , (3.2) A = (cid:18) κ κ (cid:19) , A ( t ) has eigenvalues tθ , tθ , det A ( x, t ) = κ κ ( x − ta )( x − ta )( x − a )( x − a ) . Note that we have the relation κ κ a a a a = θ θ . Define λ , µ , µ by a ( λ ) = 0 , µ = a ( λ ) /κ , µ = a ( λ ) /κ (3.3)so that µ µ = ( λ − ta )( λ − ta )( λ − a )( λ − a ) (see Eq.(14) in [3]) and introduce µ by µ = ( λ − ta )( λ − ta ) / ( qκ µ ). Then the matrix elements can be parametrizedby these variables and the gauge freedom w (see page 149 of [3]). EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 19 The first component of Y ( x ) satisfies the following equation: Y ( q x ) − (cid:16) a ( qx ) + a ( qx ) a ( x ) a ( x ) (cid:17) Y ( qx )(3.4) + a ( qx ) a ( x ) ( a ( x ) a ( x ) − a ( x ) a ( x )) Y ( x ) = 0 . In our parametrisation, we have a ( qx ) a ( x ) ( a ( x ) a ( x ) − a ( x ) a ( x )) = qx − λx − λ κ κ ( x − ta )( x − ta )( x − a )( x − a ) , (3.5) a ( qx ) + a ( qx ) a ( x ) a ( x ) = q ( qκ + κ ) x + c x + c x − λt ( θ + θ ) x − λ ,c = q κ κ ( λ − a )( λ − a ) µλ − ( q + 1)( qκ + κ ) λ − qt ( θ + θ ) λ + ( λ − a t )( λ − a t ) λµ ,c = − qκ κ ( λ − a )( λ − a ) µ + ( qκ + κ ) λ + ( q + 1) t ( θ + θ ) − ( λ − a t )( λ − a t ) µ . Note that there are two accessory parameters λ and µ , which play the role of de-pendent variables in the q -Painlev´e VI equation (see Eqs.(19), (20) in [3]). We mayregard Eq.(3.4) with Eq.(3.5) as a q -difference analogue of Eq.(1.5) in the differentialequations. We impose a restriction on the accessory parameters as we have done forEq.(1.5) to obtain the Heun equation. Here we require λ = a . Then we have Y ( q x ) − { q ( qκ + κ ) x + d x + t ( θ + θ ) } Y ( qx )(3.6) + κ κ ( qx − a )( x − ta )( x − ta )( x − a ) Y ( x ) = 0 , where d = ( a t − a )( a t − a ) a µ − a ( qκ + κ ) − qt ( θ + θ ) a . (3.7)Let u ( x ) be the function which satisfies u ( qx ) = ( x − ta )( x − ta ) u ( x ), e.g. u ( x ) = x (log t a a ) / log q ( x/ ( ta ); q ) ∞ ( x/ ( ta ); q ) ∞ . Then the function f ( x ) = Y ( qx ) /u ( qx )satisfies ( x − ta )( x − ta ) f ( qx ) − { (( qκ + κ ) /q ) x + ( d /q ) x + t ( θ + θ ) } f ( x )(3.8) + ( κ κ /q )( x − a )( x − qa ) f ( x/q ) = 0 . Note that there is the relation κ κ a a a a = θ θ . In Eq.(2.62), we set l = a tq / , l = a tq / , h = a q − / , h = a q / , (3.9) l = 1 /κ , l = q/κ , h / = θ ( a a a a κ κ ) − / , E = d / ( κ κ ) . Then we have h − / = θ ( a a a a κ κ ) − / and Eq.(3.8).Hence Eq.(3.8) is obtained by the fourth degeneration of the Ruijsenaars-van Diejenoperator. We may regard d as an accessory parameter. Linear q -difference equation related with q -Painlev´e equation of type E (1)6 . Yamada [20] derived a q -difference Painlev´e equation of type E (1)6 by Lax for-malism, i.e. the compatibility condition for two linear q -difference equations. One ofthe linear difference equations is written as Eq.(40) in [20], i.e.( b q − z )( b q − z )( b q − z )( b q − z ) t q ( f q − z ) z (cid:20) y ( z/q ) − gzt ( gz − q ) y ( z ) (cid:21) (3.10)+ ( b t − z )( b t − z )( f − z ) z t (cid:20) y ( qz ) − ( gz − t gz y ( z ) (cid:21) + (cid:20) ( b g − b g − b g − b g − t g ( f g − z ( gz − q ) − b b ( b g − t )( b g − t ) f gz (cid:21) y ( z ) = 0 . We may regard f , g as accessory parameters. The other linear equation (deformationequation) contains the difference on the variable t as well as the variable x (see Eq.(40)in [20]). By compatibility condition for two linear difference equations, we have q -Painlev´e equation of type E (1)6 for the dependent variables f and g and independentvariable t (see Eq.(39) in [20]).We now specialize the parameters f and g to f = b in Eq.(3.10). Then we obtain( b q − z )( b q − z )( b q − z ) t qz y ( z/q ) + c ( z ) z q / ( b − z ) y ( z ) + ( z − b t )( z − b t )( b − z ) z t y ( qz ) = 0 , (3.11) c ( z ) = − ( q / + q − / ) z + ( b q − / + b q / + b q / + b q / + b tq / + b tq / ) z + c + q / tb b ( b + b ) z , and the term c contains the accessory parameter g . Note that there is a relation qb b b b = b b b b in Yamada’s paper. We apply a gauge transformation by y ( z ) = z − / t ) / log q ˜ y ( z ). Then we have( z − b )( z − b q )( z − b q )( z − b q ) z ˜ y ( z/q ) + c ( z )˜ y ( z ) + ( z − b t )( z − b t )˜ y ( qz ) = 0 . (3.12)By replacing q to q − , it turns out that Eq.(3.12) is the third degeneration of theRuijsenaars-van Diejen operator in Eq.(2.46). The eigenvalue E in Eq.(2.46) essen-tially corresponds to the accessory parameter g in c .3.3. Linear q -difference equation related with q -Painlev´e equation of type E (1)7 . Yamada [20] also derived a q -difference Painlev´e equation of type E (1)7 by Laxformalism, i.e. the compatibility condition for two linear q -difference equations. Set(3.13) B ( z ) = (1 − b z )(1 − b z )(1 − b z )(1 − b z ) , B ( z ) = (1 − b z )(1 − b z )(1 − b z )(1 − b z ) . EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 21 Then one of the q -difference equations is written as Eq.(37) in [20], i.e. B ( t/z ) t ( f − z ) (cid:20) y ( qz ) − t (1 − gz ) t − gz y ( z ) (cid:21) + t B ( q/z ) q ( f q − z ) (cid:20) y ( z/q ) − qt − gzt ( q − gz ) y ( z ) (cid:21) (3.14) + (1 − t ) gz (cid:20) qB ( g )( f g − gz − q ) − t B ( g/t )( f g − t )( gz − t ) (cid:21) y ( z ) = 0 . We also obtain the q -Painlev´e equation of type E (1)7 (see Eq.(36) in [20]) by a com-patibility condition of Eq.(3.14) with the deformation equation written as Eq.(38) in[20].By specializing to f = b in Eq.(3.14) and applying a gauge transformation, wehave ( z − b t )( z − b t )( z − b t )( z − b t ) y ( qz ) − c ( z ) y ( z )(3.15) + ( z − b )( z − b q )( z − b q )( z − b q ) y ( z/q ) = 0 , where c ( z ) = q − / { (1 + q ) z + c z + c z + c z + ( b b b b + q b b b b ) t q } , (3.16) c = − ( b + b q + b q + b q + b tq + b tq + b tq + b tq ) ,c = − q ( b b b t q + b b b t q + b b b t q + b b b t q + b b b t + b b b t + b b b t + b b b t ) , and the term c contains the accessory parameter g . There is a relation qb b b b = b b b b in Yamada’s paper. Eq.(3.15) is the second degeneration of Ruijsenaars-vanDiejen operator in Eq.(2.38) by setting h = b q − / , h = b q / , h = b q / , h = b q / , (3.17) l = b q / t, l = b q / t, l = b q / t, l = b q / t. Note that we used the relation ( h h h h l l l l ) / = qt b b b b = q t b b b b ,which follows from qb b b b = b b b b .4. Degeneration of Ruijsenaars-van Diejen operator with N variables We describe Ruijsenaars-van Diejen operator (1.10) of N variables and investigatedegenerations of the opetator. By using the notation in section 2, the Ruijsenaars-vanDiejen operator of N variables is given by(4.1) A + ( h ; z ) = N X j =1 ( V j, + ( h ; z ) exp( − ia − ∂ z j ) + V j, + ( h ; − z ) exp( ia − ∂ z j )) + V b, + ( h ; z ) , where V j, + ( h ; z ) = Q n =1 R + ( z j − h n − ia − / R + (2 z j + ia + / R + (2 z j − ia − + ia + / · (4.2) · Y k = j R + ( z j − z k − µ + ia + / R + ( z j + z k − µ + ia + / R + ( z j − z k + ia + / R + ( z j + z k + ia + / ,V b, + ( h ; z ) = P t =0 p t, + ( h )[ Q Nj =1 E t, + ( µ ; z j ) − E t, + ( µ ; ω t, + ) N ]2 R + ( µ − ia + / R + ( µ − ia − − ia + / . We set h n = ˜ h n − ia + / 2. By the limit q + → 0, we have the following proposition,whose proof is similar to the case of one variable. Proposition 4.1. As q + → , the Ruijsenaars-van Diejen operator in Eq.(4.1) cor-responds to the operator (4.3) A h i ( h ; z ) = N X j =1 ( V h i j ( h ; z ) exp( − ia − ∂ z j )+ V h i j ( h ; − z ) exp( ia − ∂ z j ))+ U h i ( h ; z ) , up to some additive constant, where V h i j ( h ; z ) = Q n =1 (1 − e − πiz j e πi ˜ h n e − πa − )(1 − e − πiz j )(1 − e − πiz j e − πa − ) Y k = j (1 − e πi ( z k − z j ) e πiµ )(1 − e − πi ( z k + z j ) e πiµ )(1 − e πi ( z k − z j ) )(1 − e − πi ( z k + z j ) ) , (4.4) U h i ( h ; z ) =(4.5) Q n =1 ( e πi ˜ h n − e πiµ − e πiµ e πa − − N Y j =1 n e πiµ + ( e πiµ − e πiµ e πa − − − e πiz j e πa − )(1 − e − πiz j e πa − ) o + Q n =1 ( e πi ˜ h n + 1)2( e πiµ − e πiµ e πa − − N Y j =1 n e πiµ + ( e πiµ − e πiµ e πa − − e πiz j e πa − )(1 + e − πiz j e πa − ) o + e − πa − e N − πiµ Y n =1 e πi ˜ h n h X n =1 ( e πi ˜ h n + e − πi ˜ h n ) N X j =1 ( e πiz j + e − πiz j ) − ( e − πa − + e πa − ) N X j =1 ( e πiz j + e − πiz j )+ ( e πiµ − e − πiµ )( e πiµ e πa − − e − πiµ e − πa − ) X ≤ j By applying the gauge transformation with respect to the function ( R − ( z ) . . . R − ( z N )) ,we arrive at the following operator(4.6) ˜ A h i ( h ; z ) = N X j =1 ( ˜ V h i j ( h ; z ) exp( − ia − ∂ z j ) + ˜ W h i j ( h ; z ) exp( ia − ∂ z j )) + U h i ( h ; z ) , where U h i ( h ; z ) was defined in Eq.(4.5) and˜ V h i j ( h ; z ) = Q n =1 (1 − e − πiz j e πi ˜ h n e − πa − ) e − πa − e − πiz j (1 − e − πiz j )(1 − e − πiz j e − πa − ) · (4.7) · Y k = j (1 − e πi ( z k − z j ) e πiµ )(1 − e − πi ( z k + z j ) e πiµ )(1 − e πi ( z k − z j ) )(1 − e − πi ( z k + z j ) ) , ˜ W h i j ( h ; z ) = Q n =1 (1 − e πiz j e πi ˜ h n e − πa − ) e − πa − e πiz j (1 − e πiz j )(1 − e πiz j e − πa − ) ·· Y k = j (1 − e − πi ( z k − z j ) e πiµ )(1 − e πi ( z k + z j ) e πiµ )(1 − e − πi ( z k − z j ) )(1 − e πi ( z k + z j ) ) . Note that this operator was essentially obtained by van Diejen [18].We apply the second degeneration. Proposition 4.2. In Eq.(4.6), we replace z by z + iR , ˜ h n ( n = 1 , , , by h n + iR , ˜ h n ( n = 5 , , , by h n − iR and take the limit R → + ∞ . Then we have the operator (4.8) A h i ( h ; z ) = N X j =1 ( V h i j ( h ; z ) exp( − ia − ∂ z j ) + W h i j ( h ; z ) exp( ia − ∂ z j )) + U h i ( h ; z ) , where V h i j ( h ; z ) = e πiz j e N − πiµ Y n =1 (1 − e − πiz j e πih n e − πa − ) Y n =5 e πih n Y k = j (1 − e πi ( z k − z j ) e πiµ )(1 − e πi ( z k − z j ) ) , (4.9) W h i j ( h ; z ) = e πa − e − πiz j Y n =5 (1 − e πiz j e πih n e − πa − ) Y k = j (1 − e − πi ( z k − z j ) e πiµ )(1 − e − πi ( z k − z j ) ) , U h i ( h ; z ) = e N − πiµ Y n =5 e πih n h(cid:16) X n =1 e πih n + X n =5 e − πih n (cid:17) e − πa − N X j =1 e πiz j − (1 + e − πa − ) N X j =1 e πiz j + e − πiµ e − πa − ( e πiµ − e πiµ e πa − − X ≤ j Proposition 4.3. In Eq.(4.10), we replace z by z − iR , h n ( n = 1 , by h n − iR , h n ( n = 3 , by h n + iR , l n ( n = 1 , , , by l n − iR and take the limit R → + ∞ . EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 25 Then we have the operator (4.13) A h i ( h, l ; z ) = N X j =1 ( V h i j ( h ; z ) exp( − ia − ∂ z j ) + W h i j ( l ; z ) exp( ia − ∂ z j )) + U h i ( h, l ; z ) , where V h i j ( h ; z ) = e πiz j Y n =1 (1 − e πih n e − πa − e − πiz j ) Y k = j (1 − e πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) , (4.14) W h i j ( l ; z ) = e πiz j Y n =1 (1 − e πil n e πa − e − πiz j ) Y k = j (1 − e − πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) ,U h i ( h, l ; z ) = (cid:16) X n =1 e πih n + X n =1 e πil n (cid:17) N X j =1 e πiz j − ( e πa − + e − πa − ) N X j =1 e πiz j (4.15) + e − πiµ e − πa − ( e πiµ − e πiµ e πa − − X ≤ j Proposition 4.4. In Eq.(4.16), we replace z by z + iR , h n ( n = 1 , , , by h n + iR , l n ( n = 1 , by l n + iR , l n ( n = 3 , by l n − iR and take the limit R → + ∞ . Thenwe have the operator (4.18) A h i ( h, l ; z ) ≡ N X j =1 ( V h i j ( h ; z ) exp( − ia − ∂ z j ) + W h i j ( l ; z ) exp( ia − ∂ z j )) + U h i ( h, l ; z ) , where V h i j ( h ; z ) = e − πa − Y n =1 (1 − e πih n e − πa − e − πiz j ) Y k = j (1 − e πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) , (4.19) W h i j ( l ; z ) = e πiz j e πi ( l + l ) 2 Y n =1 (1 − e πil n e πa − e − πiz j ) Y k = j (1 − e − πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) ,U h i ( h, l ; z ) =( e πil + e πil ) N X j =1 e πiz j + e πi ( h + h ) ( e πi ( h − h ) + e πi ( h − h ) ) Y n =1 e πil n N X j =1 e − πiz j . By a multiplication and a gauge transformation we have˜ A h i ( h, l ; z ) = N X j =1 ( ˜ V h i j ( h ; z ) exp( − ia − ∂ z j ) + ˜ W h i j ( l ; z ) exp( ia − ∂ z j )) − U h i ( h, l ; z ) , (4.20)where˜ V h i j ( h ; z ) = e πiz j Y n =1 (1 − e πih n e − πa − e − πiz j ) Y k = j (1 − e πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) , (4.21)˜ W h i j ( l ; z ) = e πi ( l + l ) e πiz j Y n =1 (1 − e πil n e πa − e − πiz j ) Y k = j (1 − e − πiµ e πi ( z k − z j ) )(1 − e πi ( z k − z j ) ) . Discussion We have found out that the degenerated Ruijsenaars-van Diejen operators of onevariable appear by specializations of the linear q -difference equations related with the q -Painlev´e equations of types D (1)5 , E (1)6 and E (1)7 . Our results should be extended tothe case of the q -Painlev´e equation of type E (1)8 and the elliptic-difference Painlev´eequation. Note that Yamada and his collaborators found Lax pairs of the q -Painlev´eequations of type E (1)8 and the elliptic-difference Painlev´e equation [19, 20, 6]. OnLax pairs of the elliptic-difference Painlev´e equation, see also the papers by Rainsand Ormerod [8, 7].We propose other related problems. Komori and Hikami proved existence of thecommuting operators for the multivariable Ruijsenaars-van Diejen operator [4]. Thecommuting operators of the multivariable degenerate operators should be clarified.It is known that the Ruijsenaars-van Diejen operator of one variable admits E symmetry [11]. A kernel function plays important roles in [11], because the Hilbert-Schmidt operator of the kernel function is used to built up Hilbert space featuresand is also used to establish the invariance of the discrete spectra under the E Weyl EGENERATIONS OF RUIJSENAARS-VAN DIEJEN OPERATOR 27 group. The symmetry of the degenerate operators should also be studied well. Inparticular, the kernel functions for the degenerate operators should be established. Acknowledgements The author is grateful to Simon Ruijsenaars for valuable comments and fruitfuldiscussions. He would like to thank the university of Leeds, where most parts of thispapar were accomplished. He was supported by JSPS KAKENHI Grant NumberJP26400122 and by Chuo University Overseas Research Program. References [1] D. Guzzetti, The elliptic representation of the general Painlev´e VI equation. Comm. PureAppl. Math. (2002), 1280–1363.[2] K. Iwasaki, H. Kimura,S. Shimomura and M. Yoshida, From Gauss to Painlev´e, Aspects ofMathematics, E16, Braunschweig: Friedr. Vieweg & Sohn, 1991.[3] M. Jimbo, H. Sakai, A q -Analog of the Sixth Painlev´e Equation. Lett. Math. Phys. (1996),145–154[4] Y. Komori and K. Hikami, Quantum integrability of the generalized elliptic Ruijsenaarsmodels, J. Phys. A (1997), 4341–4364.[5] Yu. I. Manin, Sixth Painlev´e equation, universal elliptic curve, and mirror of P . 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Takemura, Heun’s differential equation (translation of Heun’s differential equation (Japan-ese), Sugaku (2008), 272–294), Selected papers on analysis and differential equations,45–68, Amer. Math. Soc. Transl. Ser. 2, , Amer. Math. Soc., Providence, 2010.[18] J. F. van Diejen, Difference Calogero-Moser systems and finite Toda chains, J. Math. Phys. (1995), 1299–1323. [19] Y. Yamada, A Lax formalism for the elliptic difference Painlev´e equation, SIGMA (2009),paper 042.[20] Y. Yamada, Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E (1) n , Int. Math. Res. Notices (2011) 3823–3838.[21] A. Zabrodin and A. Zotov, Quantum Painlev´e-Calogero correspondence for Painlev´e VI. J.Math. Phys. (2012), 073508, 19 pp. Department of Mathematics, Faculty of Science and Engineering, Chuo Univer-sity, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan E-mail address ::