Isomonodromy sets of accessory parameters for Heun class equations
aa r X i v : . [ m a t h . C A ] J a n Isomonodromy sets of accessory parameters for Heun classequations
Jun Xia ∗ , Shuai-Xia Xu † and Yu-Qiu Zhao ∗ Abstract
In this paper, we consider the monodromy and, in particularly, the isomonodromy setsof accessory parameters for the Heun class equations. We show that the Heun class equa-tions can be obtained as limits of the linear systems associated with the Painlev´e equationswhen the Painlev´e transcendents go to one of the actual singular points of the linear sys-tems. While the isomonodromy sets of accessory parameters for the Heun class equationsare described by the Taylor or Laurent coefficients of the corresponding Painlev´e functions,or the associated tau functions, at the positions of the critical values. As an applicationof these results, we derive some asymptotic approximations for the isomonodromy sets ofaccessory parameters in the Heun class equations, including the confluent Heun equation,the doubly-confluent Heun equation and the reduced biconfluent Heun equation.
Keywords and phrases:
Heun class equations, Painlev´e equations, accessory parameter,monodromy, isomonodromy deformation, asymptotic analysis.
Contents ∗ Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, China. † Institut Franco-Chinois de l’Energie Nucl´eaire, Sun Yat-sen University, GuangZhou 510275, China. Accessory parameters of DHE 15
The Heun equation (HE) is the general second-order linear ODE having four regular singularpoints, with the canonical form [25, Eq. (31.2.1)]d w d z + (cid:18) γz + δz − ǫz − a (cid:19) d w d z + αβz − qz ( z − z − a ) w = 0 , (1.1)where α + β + 1 = γ + δ + ǫ . The parameters α, β, γ, δ, ǫ determine the characteristic exponentsof the regular singularities at z = 0 , , a, ∞ : the exponents are { , − γ } , { , − δ } , { , − ǫ } and { α, β } , respectively. While the remaining parameters a and q , known as the accessoryparameters, involve global monodromy properties of (1.1).The same as the classical Gauss hypergeometric equation, the Heun equation has severalconfluent forms. Indeed, there are four standard confluent forms when two or more singularitiesmerge into one or more irregular singularities (cf. [25, Eqs. (31.12.1)-(31.12.4)]).( i ) Confluent (or, singly-confluent) Heun equation (CHE):d w d z + (cid:18) γz + δz − a (cid:19) d w d z + pz − qz ( z − a ) w = 0 . (1.2)This equation has two regular singularities, and an irregular singularity of rank 1 at infinityarising from the coalescing of two regular singularities.( ii ) Doubly-confluent Heun equation (DHE):d w d z + (cid:18)
12 + γz − a z (cid:19) d w d z + pz − qz w = 0 . (1.3)2his equation has two irregular singularities of rank 1 at zero and infinity, each originating fromthe confluence of two regular singularities.( iii ) Biconfluent Heun equation (BHE):d w d z + (cid:16) z + 2 a + γz (cid:17) d w d z + pz − qz w = 0 . (1.4)This equation possesses a regular singularity, and an irregular singularity of rank 2 at infinityarising from the coalescing of three regular singularities.( iv ) Triconfluent Heun equation (THE):d w d z + (2 z + a ) d w d z + ( pz − q ) w = 0 . (1.5)This equation has one irregular singularity of rank 3 at infinity, resulting from the coalescing ofall of the four singularities.Modified or reduced forms of the confluent Heun equations are also available. Five reducedconfluent equations appear as a result of weak confluence processes. For instance, we have thereduced triconfluent Heun equation (RTHE)d w d z − (cid:0) z + 2 az + q (cid:1) w = 0 , (1.6)which has one irregular singularity of rank 5 / w d z + 2 αz d w d z − (cid:16) z + a + qz (cid:17) w = 0 (1.7)with one irregular singularity of rank 3 / y xx = F ( x, y, y x )with F meromorphic in x and rational in y and y x , whose solutions have the Painlev´e property,that is, the only movable singularities of their solutions are poles or isolated essential singular-ities; see [16, 25]. The Painlev´e equations find important applications in various fields, such asnumber theory, statistical mechanics, random matrix theory, orthogonal polynomials, quantumgravity and quantum field theory, nonlinear optics and fibre optics, etc.; see [5, 12, 13, 25] andthe references therein.In [14, 15], Fuchs discovered a remarkable connection between the HE and the PVI. He addedan extra apparent singularity at z = y to the HE (1.1). The apparent singularity is presentedin the equation but is absent in the solution. The position of the apparent singularity y isdeformed with the regular singularity a in (1.1). Then, the PVI equation arose as a compatibilitycondition, of the deformed HE coupled with another linear equation with differentiation in the3ariable a . Inspired by the works of Fuchs, Slavyanov et al. studied the deformation of theHeun class of equations by adding an apparent singularity. It was shown that each Painlev´eequation can be considered as the isomonodromy deformation condition for a deformed equationof the Heun class; see [30]-[32]. In [29], Slavyanov also discovered that the Heun class equationscan be regarded as the quantization of the classical Hamiltonian of the Painlev´e equations. Inthis sense, there exists the following correspondence between Heun class equations and Painlev´eequations:HE → PVI , CHE → PV , BHE → PIV , DHE → PIII , THE → PII , RTHE → PI . (1.8)Here ‘ → ’ means that given an equation of Heun class there is a Painlev´e equation correspondingto it.It is well-known that every Painlev´e equation can be obtained as the compatibility conditionof a 2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) , (1.9)and ∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) . (1.10)The first-order matrix equation (1.9) is equivalent to two second-order linear ODEs satisfiedrespectively by the elements of the first row and second row of Φ( z, x ). More precisely, if wedenote the coefficient matrix A ( z, x ) by A ( z, x ) := A = (cid:18) A A A A (cid:19) , then each element of the first row of Φ( z, x ) satisfies the ODE:d Φ d z − (cid:18) Tr A + A ′ A (cid:19) dΦ d z + (cid:18) det A − A ′ + A A ′ A (cid:19) Φ = 0 , (1.11)while the elements of the second row solve a similar ODE:d Φ d z − (cid:18) Tr A + A ′ A (cid:19) dΦ d z + (cid:18) det A − A ′ + A A ′ A (cid:19) Φ = 0 . (1.12)Here, the prime indicates the differentiation with respect to z .It was observed in [12, p.86] that the linear equation (1.11) for PVI is equivalent to HE whenthe solutions to PVI approach the critical values 0 , , x, ∞ , which are the actual singular pointsof (1.11). Recently, Dubrovin and Kapaev [10] studied in detail the equivalence of HE andthe Lax pair for PVI at the movable poles of the solutions of PVI. Moreover, they establisheda connection between the accessory parameter of HE and the free parameter of the Laurentexpansion of PVI. Similar results were also derived earlier in [22]. In [6, 7], the equivalenceof CHE and the Lax pair for PV at certain critical value of PV was shown and the accessoryparameter was expressed in terms of the τ -function of PV. An application to black holes wasalso addressed therein.In this paper, we consider the monodromy and isomonodromy deformation of the Heunclass equations. Firstly, we describe the monodromy of Heun class equations and considerthe isomonodromy deformation by using the linear systems for the corresponding Painlev´eequations. We show that the Heun class equations can be obtained as limits of the linear system41.11) or (1.12) associated with the Painlev´e equations I-VI and XXXIV as the correspondingPainlev´e transcendents y ( x ) approach one of the actual singular points of the linear system:PI → RTHE , PII → THE , PXXXIV → RBHE , PIII → DHE , (1.13)and PIV → BHE , PV → CHE , PVI → HE . (1.14)Moreover, the accessory parameters in these equations of Heun class are determined by thecorresponding Painlev´e functions and τ -functions. Secondly, using the the limiting procedurein (1.13), (1.14) and the monodromy of a specific Heun class equation, we show that there isa discrete set of pairs of accessory parameters ( a n , q n ), such that the equation of Heun classcorresponding to these parameters has the same monodromy data. Under a bijection, thediscrete set coincides with the set of parameters ( a n , b n ) in the Taylor or Laurent expansions,respectively at the zeros or poles a n , of the unique solution to the corresponding Painlev´eequation with the same monodromy data as the Heun class equation. Finally, using knownasymptotic expansions for the Painlev´e transcendents and the associated τ -functions in theliterature, we derive some asymptotic approximations for the isomonodromy sets of accessoryparameters in the Heun class equations.The rest of this paper is organized in the following way. We consider the monodromydata and isomonodromy deformation of the Heun class equations RTHE-HE in Sections 2-8, respectively. We describe the isomonodromy sets of accessary parameters by the sets ofparameters in the Taylor or Laurent expansion near the zeros or poles of the solution of thecorresponding Painlev´e equation with the same monodromy data as the Heun class equation.The main results are stated in Theorems 1-7 at the end of each section. The proofs of Theorem1 and Theorem 7 are given in Section 2 and Section 8, respectively, with details included.These proofs are concerned with two representative examples of Heun class equations, namelythe HE with four regular singularities and the RTHE with an irregular singularity. WhileTheorems 2-6 can be proved in the same manner and we skip the detail. In the last section,we derive asymptotic approximations for isomonodromy sets of accessory parameters of someHeun class equations, expressed in terms of the monodromy data. The equations of Heun classwe considered in this section include the RBHE, CHE, and DHE. Consider the RTHE equation (1.6) with parameters a and q . There exist unique solutions, ofthe form Y k ( z ) = ( y k ( z ) , y k ( z )), which satisfy the normalized asymptotic behavior as z → ∞ Y k ( z ) ∼ z − (1 , e ( z + az ) σ , z ∈ Ω k , (2.1)where Ω k , k = − , · · · ,
4, are the Stokes sectors defined byΩ k = n z ∈ C : π k − < arg z < π k − o , (2.2)and σ is one of the Pauli matrices, σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) and σ = (cid:18) − (cid:19) . Y k +1 ( z ) = Y k ( z ) S k , k = − , · · · , , (2.3)where S k − = (cid:18) s k − (cid:19) , S k = (cid:18) s k (cid:19) . (2.4)Moreover, it follows from the asymptotic behavior in (2.1) that Y ( e πi z ) = iY − ( z ) σ . (2.5)Using (2.3)-(2.5), we find the cyclic condition S − · · · S = iσ . (2.6)The restriction (2.6) contributes four scalar algebraic equations, three among them are inde-pendent. Accordingly, the Stokes multipliers s k fulfill s k = i (1 + s k +2 s k +3 ) (2.7)for all integer k , with s k +5 = s k . The monodromy data is constituted by the set of Stokesmultipliers, which is a 2-dimensional surface in C described by (2.7) { ( s , s , s , s , s ) ∈ C : s , . . . s satisfy (2.7) } . (2.8) To study the isomonodromy deformation of the RTHE equation, it is convenient to consider a2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , (2.9)where A ( z, x ) = (cid:18) − v z + yz + y + x z − y ) v (cid:19) , B ( z, x ) = (cid:18) z + y (cid:19) . The compatibility condition of the Lax pair reads d y d x = v, d v d x = 6 y + x, (2.10)which leads to the PI equation d y d x = 6 y + x. The τ -function associated with PI is defined by (see [20, (C.7)]) σ ( x ) = dd x log τ ( x ) = 12 v − y − xy. (2.11)6hen, we have (see [20, (C.9)]) (cid:18) d σ d x (cid:19) + 4 (cid:18) d σ d x (cid:19) + 2 x d σ d x − σ = 0 . (2.12)There exist unique solutions Φ k ( z, x ) of the equation (2.9), which satisfy the normalizedasymptotic behavior as z → ∞ Φ k ( z, x ) = (cid:16) z (cid:17) σ σ + σ i √ (cid:16) I + O (cid:16) z − (cid:17)(cid:17) e (cid:16) z + xz (cid:17) σ , z ∈ Ω k , (2.13)where Ω k for k = − , · · · , k +1 ( z ) = Φ k ( z ) ˆ S k , k = − , · · · , , (2.14)where ˆ S k − = (cid:18) s k − (cid:19) , ˆ S k = (cid:18) s k (cid:19) . (2.15)The behavior (2.13) also gives Φ ( e πi z ) = i Φ − ( z ) σ . (2.16)From (2.14)-(2.16), we find the cyclic conditionˆ S − ˆ S · · · ˆ S = iσ . (2.17)This condition implies the following algebraic equationsˆ s k = i (1 + ˆ s k +2 ˆ s k +3 ) , (2.18)with k ∈ Z and ˆ s k +5 = ˆ s k . Thus, the Stokes multipliers are described by the 2-dimensionalsurface (2.8) in C with the coordinates (ˆ s k ) ≤ k ≤ . In this subsection, we show that the RBHE can be obtained as a limit of the linear system(1.12) associated with the Lax pair for the Painlev´e I equation when x tends to one of the polesof the Painlev´e I transcendents.Substituting the enties of A ( z, x ) into (1.12), we obtaind Φ d z − z − y dΦ d z − Q ( z, x )Φ = 0 , (2.19)where Q ( z, x ) = 4 z + 2 xz + v − y − xy − vz − y = 4 z + 2 xz + 2 dd x log τ ( x ) − vz − y . To obtain RTHE, we need to eliminate the singularity z = y in (2.19). This can be realizedby considering the poles of y . It is known that y admits the following Laurent expansion neara pole (see [16, Eq. (1.1)] ): y ( x ) = 1( x − a ) − a
10 ( x − a ) −
16 ( x − a ) + b ( x − a ) + O (( x − a ) ) , (2.20)7here b is an arbitrary parameter.Using the first equation of (2.10) to replace v by y ′ and substituting (2.20) into the expressionof Q in (2.19), we obtain the RTHE as x → a d Φ d z − (cid:0) z + 2 az + q (cid:1) Φ = 0 , (2.21)with the accessory parameter q given by q = lim x → a (cid:18) dd x log τ ( x ) − x − a (cid:19) = − b. (2.22) As shown in Section 2.3, the isomonodromy deformation of (2.9) is described by the PI equation.While the RTHE can be obtained as a limit of the second row of the isomonodromy family ofΦ( z, x ), when x tends to one of the pole of the Painlev´e I transcendent. The accessory parametersof RTHE are then expressed in terms of the parameters in the Laurent expansion near the polesof the PI transcendents. As a consequence, we obtain a family of accessory parameters such thatthe corresponding RTHE shares the same monodromy data, as stated in the following theorem. Theorem 1.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that the RTHE (1.6) corresponding to these parameters has the same monodromy data (2.8) as the original onewith the parameters a and q . This set coincides, via the transformation q n = − b n , with the setof parameters ( a n , b n ) in the Laurent expansion (2.20) near the poles a n , of the unique solutionof PI corresponding to the same monodromy data (2.8) as the RTHE.Proof. Consider the RTHE (1.6) with given accessory parameters a and q . There exist uniquesolutions of (1.6) which satisfy the normalized asymptotic behavior (2.1) as z → ∞ . Thesolutions are related by the Stokes matrices S k as given in (2.3) and (2.4), while the independentStokes multipliers constitute the monodromy data (2.8) for the RTHE. Moreover, there existunique solutions Φ k ( z, x ) of the system (2.9) with the normalized behavior (2.13) at infinity,where the x -dependence of Φ k ( z, x ) is described by the unique solution of the PI equationhaving the Laurent expansion (2.20). Here we let a be a pole of the PI solution and choose theparameter b according to (2.22), namely b = − q . The solutions Φ k ( z, x ) are related by theStokes matrices ˆ S k defined in (2.14) and (2.15), which are independent of x and have the sametriangular-matrix forms as the Stokes matrices S k for the RTHE.From the limiting procedure demonstrated in Section 2.3, we obtain the RTHE equationwith accessory parameters a and q , as a limit of the second row of the isomonodromy familyΦ k ( z ; x ) when x → a . It follows from the x -independence of the monodromy data that theStokes matrices S k = ˆ S k . Thus, we have shown that any given accessory parameters ( a, q ) isrelated, via q = − b , to the pole parameters ( a, b ) of the unique solution of PI correspondingto the same monodromy data (2.8). It is known that each Painlev´e tanscendents has infinitymany poles which are discrete in the complex plane. Therefore the set of pairs of accessoryparameters of RTHE sharing the same monodromy data are also discrete. Thus, we completethe proof of Theorem 1. 8 Accessory parameters of THE
Consider the THE (1.5) with the accessory parameters a and q and the fixed parameter p .There exist unique solutions of (1.5) with normalized asymptotic behavior as z → ∞ Y k ( z ) ∼ e − ( z + a z ) z − (1 , e ( z + a z ) σ z ( µ − ) σ , z ∈ Ω k , (3.1)where the Stokes sectors are defined byΩ k = n z ∈ C : π k − < arg z < π k + 1) o , (3.2)for k = 1 , · · · ,
7. Here the parameter µ is given by µ = (3 − p ) /
2. The solutions are related bythe Stokes matrices Y k +1 ( z ) = Y k ( z ) S k , k = 1 , . . . , , (3.3)where S k − = (cid:18) s k − (cid:19) , S k = (cid:18) s k (cid:19) . (3.4)The behavior (3.1) also gives Y ( e πi z ) = Y ( z ) e πi (2 µ − σ . (3.5)We define the monodromy data of THE by { S k } k =1 . (3.6)From (3.3)-(3.5), we see that the Stokes matrices fulfill the cyclic condition S S · · · S = e πi (2 µ − σ . (3.7)Thus, the Stokes multipliers s k , k = 1 , · · · , s s + e πiµ (1 + s s ) = 0 , s s + e − πiµ (1 + s s ) = 0 ,s + s + s s s − e πiµ s = 0 . (3.8)These equations describe a 3-dimensional surface in C with the coordinates ( s k ) ≤ k ≤ . To study the isomonodromy deformation of THE, it is convenient to consider a 2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , (3.9)9here A ( z, x ) = (cid:18) z + v + x u ( z − y ) − u (cid:0) vz + vy + − µ (cid:1) − z − v − x (cid:19) , B ( z, x ) = (cid:18) z u − vu − z (cid:19) . The compatibility condition of the above Lax pair gives d y d x = y + v + x , d v d x = − yv −
12 + µ, d u d x = − uy, (3.10)which yields the PII equation d y d x = 2 y + xy + µ. (3.11)The PII τ -function is defined by (see [20, (C.15)])dd x log τ ( x ) = 12 v + (cid:16) y + x (cid:17) v + (cid:18) − µ (cid:19) y. (3.12)There exist unique solutions Φ k ( z, x ) of the equation (3.9) specified by the normalized asymp-totic behavior as z → ∞ Φ k ( z, x ) = ( I + O (1 /z )) e ( z + xz ) σ z ( µ − ) σ , z ∈ Ω k , (3.13)where Ω k for k = 1 , · · · , k +1 ( z ) = Φ k ( z ) ˆ S k , k = 1 , · · · , , (3.14)where ˆ S k − = (cid:18) s k − (cid:19) , ˆ S k = (cid:18) s k (cid:19) . (3.15)We define the monodromy data of the system (3.9) by { ˆ S k } k =1 ,..., . (3.16)The behavior (3.13) also implies Φ ( e πi z ) = Φ ( z ) e πi (2 µ − σ . (3.17)From (3.14)-(3.17), we find the cyclic conditionˆ S ˆ S · · · ˆ S = e πi (2 µ − σ . (3.18)The equation gives us three independent equations of the Stokes multipliers ˆ s k , k = 1 , · · · , .3 Reduction of the linear system for PII to THE In this subsection, we derive THE from the linear system (1.11) for PII at each pole of thesolutions of PII.Substituting A ( z, x ) into (1.11) givesd Φ d z − z − y dΦ d z − Q ( z, x )Φ = 0 , (3.19)where Q ( z, x ) = z + xz + (1 + 2 µ ) z + x v + (cid:0) y + x (cid:1) v + (1 − µ ) y − z + x + vz − y = z + xz + (1 + 2 µ ) z + x x log τ ( x ) − z + x + vz − y . To eliminate the singularity z = y in (3.19), we consider the poles of y . Let x = a be a poleof y , then y possesses the following Laurent expansion (see [16, (17.1)]): y ( x ) = εx − a − εa x − a ) − µ + ε x − a ) + b ( x − a ) + O (( x − a ) ) , (3.20)where ε = ε ± = ± b ∈ C is arbitrary. From the first equation of the compatibilitycondition in (3.10) and (3.20), we obtain v ( x ) = − x − a ) + O (1) , ε = ε + ,O ( x − a ) , ε = ε − . (3.21)Hence, taking x → a in (3.19) and then making the transformationΦ = w exp (cid:18) z + a z (cid:19) , we obtain the THEs d w d z + (2 z + a ) d w d z + ( pz − q ) w = 0 ,p = 3 − µ,q = lim x → a (cid:18) dd x log τ ( x ) − x − a (cid:19) = − a − b, (3.22)for ε = ε + = 1, d w d z + (2 z + a ) d w d z + ( pz − q ) w = 0 ,p = 1 − µ,q = lim x → a x log τ ( x ) = − a
18 + 10 b, (3.23)for ε = ε − = −
1. 11 .4 Isomonodromy set of accessory parameters of THE
Thus, we see that THE can be obtained as a limit of the first row of the isomonodromy familyof Φ( z, x ), when x → a , with a being the pole of the Painlev´e II transcendent correspondingto the same monodromy data of THE. While, the accessory parameters ( a, q ) are expressed interms of the parameters in the Laurent expansion of the PII transcendent. Consequently, wehave the following description of the isomonodromy set of accessory parameters of THE. Theorem 2.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that the THE (1.5) , corresponding to these parameters has the same monodromy data (3.6) and (3.8) as theoriginal equation with the parameters a and q . Under the bijection given in the third equation of (3.22) , this set coincides with the set of parameters ( a n , b n ) in the Laurent expansion (cf. (3.20) )near the poles a n of the unique solution of PII equation (3.11) with the parameter µ = (3 − p ) / and the same monodromy data (3.6) and (3.8) as the THE. Consider the RBHE (1.7) with the fixed parameter 2 α and the accessory parameters a and q .There exist two linear independent solutions Y k = ( y k , y k ) of (1.7) normalizing the asymptoticbehavior as z → ∞ Y k ( z ) ∼ z − ( α + ) (1 , i ) e − ( z + az ) σ , z ∈ Ω k , (4.1)where Ω k for k = − , · · · , k = n z ∈ C : π k − < arg z < π k + 1) o . (4.2)The solutions are related to each other by the Stokes matrices Y k +1 ( z ) = Y k ( z ) S k , k = − , , , (4.3)where S ± = (cid:18) s ± (cid:19) , S = (cid:18) s (cid:19) . (4.4)The behavior (4.1) also gives Y ( e πi z ) = e − πiα Y − ( z ) iσ . (4.5)Moreover, from (1.7) we see that the characteristic exponents at z = 0 are { , − α } . Hence,as z →
0, we have the asymptotic behavior Y ( z ) ∼ z − α (1 , z ) z ασ E (4.6)for 2 α Z , with E being some invertible constant matrix. Using (4.3)-(4.6), we obtain thesame cyclic condition as that for the Lax pair of the PXXXIV equation S − S S = iσ E − e − πiασ E. (4.7)This condition implies the following algebraic equation s − s s + s − + s − s = − πα ) . (4.8)Thus, the monodromy data for RBHE (1.7) are described by the 2-dimensional complex surface (cid:8) ( s − , s , s ) ∈ C : s − s s + s − + s − s = − πα ) (cid:9) . (4.9)12 .2 Isomonodromy deformation and PXXXIV equation To study the isomonodromy deformation of the RBHE (1.7), we regard z α Y k ( z ) as the first rowof a 2 × k ( z, x ) when x → a . Then, we look for the system Φ k ( z, x ) such thatΦ k ( z, x ) satisfy the same jump relations (4.3)-(4.5) and the following asymptotic behaviorsΦ k ( z, x ) = z − σ I + iσ √ (cid:16) I + O (cid:16) z − (cid:17)(cid:17) e − (cid:16) z + xz (cid:17) σ , z ∈ Ω k , (4.10)as z → ∞ , where Ω k , k = − , · · · ,
2, are the Stokes sectors defined in (4.2), andΦ ( z, x ) = Φ (0) ( x ) ( I + O ( z )) z ασ E, z → , (4.11)where the parameter 2 α Z and Φ (0) ( x ) is a matrix independent of z . The behavior of Φ k ( z, x )as z → k ( z, x ) satisfies the following Lax pair with rational coefficients (see[12, Chapter 5], [18, Lemma 3.2] and [34]) ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , (4.12)where A ( z, x ) = y ′ z i − i yz − iz − i ( y + x ) − i ( y ′ ) − (2 α ) yz − y ′ z ! , B ( z, x ) = (cid:18) i − iz − i ( y + x ) 0 (cid:19) . The Lax pair is related by a gauge transformation to the Lax pair for PII found by Flaschkaand Newell; see for instance [12, Chapter 5] and [18, Equation (3.19)].The compatibility condition of the above Lax pair is described by the PXXXIV equationd y d x = 12 y (cid:18) d y d x (cid:19) + 4 y + 2 xy − (2 α ) y . (4.13)Let v ( x ) = y ′ ( x ) − α y ( x ) , then − − / v ( − − / x ) satisfies the PII equation (3.11) with µ = − (2 α + ). The associated τ -function can be defined asdd x log τ ( x ) = − y ( x ) + (cid:0) v ( x ) − x (cid:1) y ( x ) + 2 αv ( x ) , (4.14)which is related simply to the standard Hamiltonian for the PII equation; cf. [25, Equation(32.6.9)].Thus, we see that the solutions Φ k ( z, x ) to the Lax pair (4.12) subject to the boundaryconditions (4.10)-(4.11) share the same monodromy data as RBHE (independent of x ) given in(4.9). While the isomonodromy deformation of Φ k ( z, x ) with respect to x are described by thePXXXIV equation.In the next subsection, we show that the RBHE can actually be obtained as a limit of thelinear system (1.11), associated with the Lax pair for the Painlev´e XXXIV equation, when thePainlev´e XXXIV transcendent y ( x ) tends to zero or infinity.13 .3 Reduction of the linear system for PXXXIV to RBHE Substituting the elements of A ( z, x ) into (1.11) givesd Φ d z − (cid:18) z − y − z (cid:19) dΦ d z − P ( z, x )Φ = 0 , (4.15)where P ( z, x ) = z + x + (( y ′ ) − α ) / (4 y ) − y − xyz + α z − y ′ z ( z − y )= z + x + 1 z (cid:18) dd x log τ ( x ) (cid:19) + α z − y ′ z ( z − y ) . To eliminate the singularity z = y , we consider the poles and zeros of y . It is readily seen that y solving (4.13) possesses the following Laurent expansion at a pole y ( x ) = 1( x − a ) − a −
12 ( x − a ) + b ( x − a ) + O (cid:0) ( x − a ) (cid:1) , (4.16)and the Taylor expansion at a zero y ( x ) = ε ( x − a ) + b ( x − a ) + 2 aε x − a ) + O (cid:0) ( x − a ) (cid:1) , (4.17)where ε = ε ± = ± α and b is arbitrary.Taking x → a in (4.15) and making the transformationΦ = z α w, (4.18)we see that w ( z ) satisfies the following RBHEs d w d z + 2 α + 1 z d w d z − (cid:16) z + a + qz (cid:17) w = 0 ,q = lim x → a (cid:18) dd x log τ ( x ) − x − a (cid:19) = a − b, (4.19)when a is a pole of y ( x ), and d w d z + 2 αz d w d z − (cid:16) z + a + qz (cid:17) w = 0 ,q = lim x → a dd x log τ ( x ) = b, (4.20)when a is a zero of y ( x ) with coefficient ε = ε + = 2 α in (4.17). When a is a zero with coefficient ε = ε − = − α , taking x → a and the transformation Φ = z α +1 w in (4.15) we arrive at d w d z + 2( α + 1) z d w d z − (cid:16) z + a + qz (cid:17) w = 0 ,q = lim x → a dd x log τ ( x ) = b. (4.21)14 .4 Isomonodromy set of accessory parameters of RBHE Thus, we see that RBHE can be obtained as a limit of the first row of the isomonodromy family ofΦ( z, x ) when x → a , a being a zero or pole of the Painlev´e XXXIV transcendents correspondingto the same monodromy data as RBHE. While the accessory parameters ( a, q ) are expressed interms of the parameters in the Taylor or Laurent expansion of the PXXXIV transcendents. Asa consequence, we have the following description of the isomonodromy accessory parameters setof RBHE. Theorem 3.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that theRBHE (1.7) corresponding to these parameters has the same monodromy data (4.9) as theoriginal equation with the parameters a and q . This set coincides with the set of parameters ( a n , b n ) in the Taylor expansion (4.17) near the zeros a n of the unique solution of the PXXXIVequation (4.13) with the same monodromy data (4.9) as the RBHE. Consider the DHE (1.3) with the accessory parameters a and q and the fixed parameters γ and p . Let Ω ( ∞ ) k , k = 1 ,
2, be the Stokes sectorsΩ ( ∞ ) k = n z ∈ C : π − k + 1) < arg z < π − k + 5) o . (5.1)There exist unique linear independent solutions of (1.3), namely Y ( ∞ ) k = ( y ( ∞ ) k , y ( ∞ ) k ), whichsatisfy the normalized asymptotic behavior as z → ∞ Y ( ∞ ) k ( z ) ∼ e − z z − θ (1 , /z ) e z σ z − θ ∞ σ , z ∈ Ω ( ∞ ) k , (5.2)for k = 1 ,
2. Here the parameters θ = γ − θ ∞ = 4 p − γ + 1. The solutions are related toeach other through the Stokes matrices Y ( ∞ )2 ( z ) = Y ( ∞ )1 ( z ) S ( ∞ )1 , S ( ∞ )1 = s ( ∞ )1 ! , (5.3) Y ( ∞ )1 ( ze πi ) e πiθ e πiθ ∞ σ = Y ( ∞ )2 ( z ) S ( ∞ )2 , S ( ∞ )2 = s ( ∞ )2 ! . (5.4)Parallelly, there exist unique linear independent solutions of (1.3) determined by the asymptoticbehavior as z → Y (0) k ( z ) ∼ e − a z z − θ (1 , z ) e a z σ z θ σ , z ∈ Ω (0) k , (5.5)where Ω (0) k for k = 1 , (0) k = n z ∈ C : π − k + 1) < arg (cid:0) a /z (cid:1) < π − k + 5) o . (5.6)The solutions are related to each other by the Stokes matrices Y (0)2 ( z ) = Y (0)1 ( z ) S (0)1 , S (0)1 = s (0)1 ! , (5.7)15 (0)1 ( ze − πi ) e − πiθ e πiθ σ = Y (0)2 ( z ) S (0)2 , S (0)2 = s (0)2 ! . (5.8)Moreover, the solutions Y ( ∞ )1 ( z ) and Y (0)2 ( z ) are connected by Y ( ∞ )1 ( z ) = Y (0)2 ( z ) E, (5.9)where E is some invertible constant matrix. We define the monodromy data of (1.3) by the set n e πiθ ∞ , e πiθ ; S ( ∞ )1 , S ( ∞ )2 ; S (0)1 , S (0)2 ; E o . (5.10)Combining (5.3), (5.4), and (5.7)-(5.9), we have the cyclic condition S ( ∞ )1 S ( ∞ )2 e − πiθ ∞ σ = E − (cid:16) S (0)1 (cid:17) − e πiθ σ (cid:16) S (0)2 (cid:17) − E. (5.11)The condition implies that the Stokes multipliers satisfy the equation e πiθ ∞ s ( ∞ )1 s ( ∞ )2 + 2 cos( πθ ∞ ) = e πiθ s (0)1 s (0)2 + 2 cos( πθ ) . (5.12) To study the isomonodromy deformation of the RTHE equation, it is convenient to consider a2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , (5.13)where A ( z, x ) = (cid:18) x − θ ∞ z + v − x z rz − uvz sz + v − x uz − x + θ ∞ z − v − x z (cid:19) ,B ( z, x ) = (cid:18) z − v − x xz rx + uvxzsx − v − x xuz − z + v − x xz (cid:19) . If we set y = − r/uv and θ vx (cid:20) θ ∞ (cid:16) − xv (cid:17) + x − vuv r + 2 us (cid:21) , then the compatibility condition of the above Lax pair reads x d y d x = (4 v − x ) y + (2 θ ∞ − y + x,x d v d x = − yv + (2 xy − θ ∞ + 1) v + 12 ( θ + θ ∞ ) x,x d u d x = u (cid:16) − x v ( θ + θ ∞ ) − xy + θ ∞ (cid:17) , (5.14)16hich implies the PIII equationd y d x = 1 y (cid:18) d y d x (cid:19) − x d y d x + 2 θ y + 2 − θ ∞ x + y − y . The PIII τ -function is defined by (see [20, (C.27)]) x dd x log τ ( x ) = 2 y v + (cid:0) − xy + 2 θ ∞ y + x (cid:1) v − ( θ + θ ∞ ) xy − x − θ − θ ∞ . (5.15)Using (5.14) and (5.15) to eliminate v ( x ), we have x dd x log τ ( x ) = x y ′ y + xy ′ y − x y − x y − θ ∞ x y − θ xy − θ + θ ∞ . (5.16)There exist unique solutions Φ ( ∞ ) k ( z, x ) of the equation (5.13), which satisfy the normalizedasymptotic behavior as z → ∞ Φ ( ∞ ) k ( z, x ) = ( I + O (1 /z )) e xz σ z − θ ∞ σ , xz ∈ Ω ( ∞ ) k , (5.17)where Ω ( ∞ ) k for k = 1 , ( ∞ )2 ( z ) = Φ ( ∞ )1 ( z ) ˆ S ( ∞ )1 , ˆ S ( ∞ )1 = s ( ∞ )1 ! , (5.18)Φ ( ∞ )1 ( ze πi ) e πiθ ∞ σ = Φ ( ∞ )2 ( z ) ˆ S ( ∞ )2 , ˆ S ( ∞ )2 = s ( ∞ )2 ! . (5.19)Similarly, there exist unique solutions Φ (0) k ( z, x ) of the equation (5.13) such thatΦ (0) k ( z, x ) = G k ( x )( I + O ( z )) e x z σ z θ σ , z ∈ Ω (0) k , z → , (5.20)where G k ( x ) are matrices independent of z , and the sectors Ω (0) k , k = 1 ,
2, are defined in (5.6)with arg( a /z ) replaced by arg( x/z ). These solutions are related to each other by the Stokesmatrices Φ (0)2 ( z ) = Φ (0)1 ( z ) ˆ S (0)1 , ˆ S (0)1 = s (0)1 ! , (5.21)Φ (0)1 ( ze − πi ) e πiθ σ = Φ (0)2 ( z ) ˆ S (0)2 , ˆ S (0)2 = s (0)2 ! . (5.22)Moreover, the solutions Φ ( ∞ )1 ( z ) and Φ (0)2 ( z ) are connected throughΦ ( ∞ )1 ( z ) = Φ (0)2 ( z ) ˆ E, (5.23)where ˆ E is some invertible constant matrix. The monodromy data of the system (5.13) isdefined as the set n e πiθ ∞ , e πiθ ; ˆ S ( ∞ )1 , ˆ S ( ∞ )2 ; ˆ S (0)1 , ˆ S (0)2 ; ˆ E o . (5.24)Applying (5.18), (5.19), and (5.21)-(5.23), we find the cyclic condition for the monodromy dataˆ S ( ∞ )1 ˆ S ( ∞ )2 e − πiθ ∞ σ = ˆ E − (cid:16) ˆ S (0)1 (cid:17) − e πiθ σ (cid:16) ˆ S (0)2 (cid:17) − ˆ E. (5.25)Thus, the Stokes multipliers satisfy the equation e πiθ ∞ ˆ s ( ∞ )1 ˆ s ( ∞ )2 + 2 cos( πθ ∞ ) = e πiθ ˆ s (0)1 ˆ s (0)2 + 2 cos( πθ ) . (5.26)17 .3 Reduction of the linear system for PIII to DHE In this section, we will show that the DHE (1.3) can be obtained as a limit of the linear system(1.11) associated with the Lax pair for the Painlev´e III equation as x → a , where a is one ofthe poles or zeros of the solution to the Painlev´e III equation.Substituting the elements of A ( z, x ) into (1.11) givesd Φ d z − (cid:18) z + 1 /y − z (cid:19) dΦ d z − Q ( z, x )Φ = 0 , (5.27)where Q ( z, x ) = x
16 + (2 − θ ∞ ) x z − θ x z + x z − z + 1 /y (cid:18) x − θ ∞ z + v − x z (cid:19) + 1 z (cid:20) y v + 12 ( − xy + 2 θ ∞ y + x ) v −
14 ( θ + θ ∞ ) xy − x θ ∞ − θ ∞ (cid:21) = x
16 + (2 − θ ∞ ) x z − θ x z + x z − z + 1 /y (cid:18) x − θ ∞ z + v − x z (cid:19) + 1 z (cid:20) x dd x log τ ( x ) + θ + θ ∞ − θ ∞ (cid:21) . To obtain DHE, we eliminate the extra singularity z = − /y in the above equation byconsidering the poles and zeros of y . It is known that y admits the following Laurent expansionnear a pole y ( x ) = εx − a − θ + ε a + b ( x − a ) + O (( x − a ) ) , (5.28)and the Taylor expansion near a zero y ( x ) = σ ( x − a ) + σ − θ ∞ a ( x − a ) + b ( x − a ) + O (( x − a ) ) , (5.29)where ε = ε ± = ± σ = σ ± = ± a = 0 and b is arbitrary.In view of the first equation of compatibility condition (5.14), (5.28) and (5.29), we obtainthat v ( x ) = O ( x − a ) , ε = ε + , a + O ( x − a ) , ε = ε − ,O (1) , σ = σ + , − a ( x − a ) − + O (1) , σ = σ − . (5.30)Thus, taking x → a in (5.27) and settingΦ = z θ e a ( z + z ) w, s = az, we obtain the DHEs d w d s + (cid:18)
12 + 1 + θ s − a s (cid:19) d w d s + ps − qs w = 0 ,p = 14 ( θ ∞ + θ ) ,q = lim x → a x dd x log τ ( x ) + a − θ − θ ∞ − a b + a θ + 1)(2 θ + 5) (5.31)18or ε = ε + , d w d s + (cid:18)
12 + 2 + θ s − a s (cid:19) d w d s + ps − qs w = 0 ,p = 14 ( θ ∞ + θ ) ,q = lim x → a x dd x log τ ( x ) + a − θ − θ ∞ − θ + θ ∞ σ = σ + , and d w d s + (cid:18)
12 + 2 + θ s − a s (cid:19) d w d s + ps − qs w = 0 ,p = 14 ( θ ∞ + θ + 2) ,q = lim x → a (cid:18) x dd x log τ ( x ) − ax − a (cid:19) + a − θ − θ ∞ − θ −
34 (5.33)for σ = σ − . For ε = ε − in (5.27), taking x → a and settingΦ = z − θ e a ( z − z ) w, s = az, we obtain the DHE d w d s + (cid:18)
12 + 1 − θ s + a s (cid:19) d w d s + ps − qs w = 0 ,p = 14 ( θ ∞ − θ ) ,q = lim x → a x dd x log τ ( x ) − a − θ − θ ∞ . (5.34) Consider the DHE (1.3) with the accessory parameters a and q and the fixed parameters γ and p . We have shown that the DHE can be obtained as a limit of the first row of the isomonodromyfamily of Φ( z, x ), when x tends to one of the poles or zeros of the Painlev´e III transcendentscorresponding to the same monodromy data of the DHE. While the accessory parameters ( a, q )are expressed in terms of the parameters in the Laurent or Taylor expansion of the PIII tran-scendents with the parameters θ = γ − θ ∞ = 4 p − γ + 1. As a result, we have thefollowing description of the isomonodromy set of accessory parameters of DHE. Theorem 4.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that the DHE (1.3) corresponding to these parameters has the same monodromy data as the original equationwith the parameters a and q . Under the bijection given in the last equation of (5.31) , this setcoincides with the set of parameters ( a n , b n ) in the Laurent expansion near the poles a n of theunique solution of PIII with the same monodromy data (5.10) as the DHE. Consider the BHE (1.4) with the accessory parameters a and q , the other fixed parameters γ and p . There exist unique solutions of (1.4) which satisfy the normalized asymptotic behavior19s z → ∞ Y k ( z ) ∼ e − z − az z − ( θ +1) (1 , e z az σ z − θ ∞ σ , z ∈ Ω k , (6.1)where the parameters θ = ( γ − / θ ∞ = ( p − γ − / k , k = 1 , , , k = n z ∈ C : π k − < arg z < π k − o . (6.2)The solutions are related to each other by the Stokes matrices Y k +1 ( z ) = Y k ( z ) S k , k = 1 , , , (6.3) Y ( z ) = Y ( ze πi ) S e πiθ e πiθ ∞ σ , (6.4)where S k − = (cid:18) s k − (cid:19) , S k = (cid:18) s k (cid:19) , k = 1 , . (6.5)The solution Y ( z ) has the asymptotic behavior as z → Y ( z ) ∼ z − θ (1 , z θ σ E, (6.6)with some invertible constant matrix E . The monodromy data of BHE (1.4) is defined as theset n e πiθ ∞ , e πiθ ; S , S , S , S ; E o . (6.7)By (6.3)-(6.6), they satisfy the cyclic condition S S S S e πiθ ∞ σ = E − e − πiθ σ E. (6.8)Thus, the Stokes multipliers s k , k = 1 , , ,
4, satisfy the algebraic equation e πiθ ∞ (1 + s s ) + e − πiθ ∞ [ s s + (1 + s s )(1 + s s )] = 2 cos(2 πθ ) . (6.9)It is direct to see that the cyclic condition (6.8) also specifies the connection matrix E to withina left-multiplicative diagonal matrix. To study the isomonodromy deformation of the BHE, it is convenient to consider a 2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , (6.10)where A ( z, x ) = z + x + θ − vz u (cid:0) − y z (cid:1) u (cid:16) v − θ − θ ∞ + v − θ vyz (cid:17) − z − x − θ − vz ! ,B ( z, x ) = (cid:18) z u u ( v − θ − θ ∞ ) − z (cid:19) . d u d x = − u ( y + 2 x ) , d y d x = − v + y + 2 xy + 4 θ , d v d x = − ( v − θ − θ ∞ ) y − v ( v − θ ) y , (6.11)which leads to the PIV equationd y d x = 12 y (cid:18) d y d x (cid:19) + 32 y + 4 xy + 2( x + 1 − θ ∞ ) y − θ y . (6.12)The PIV τ -function is defined by (see [20, (C.35)])dd x log τ ( x ) = 2 y v − (cid:18) y + 2 x + 4 θ y (cid:19) v + ( θ + θ ∞ )( y + 2 x ) . (6.13)There exist unique solutions Φ k ( z, x ) of the equation (6.10) subject to the normalized asymp-totic behavior as z → ∞ Φ k ( z, x ) = ( I + O (1 /z )) e z xz σ z − θ ∞ σ , z ∈ Ω k , (6.14)where Ω k , k = 1 , , ,
4, are the Stokes sectors defined in (6.2). Using (6.14), it is seen that thesolutions are related to each other by the Stokes matricesΦ k +1 ( z ) = Φ k ( z ) ˆ S k , k = 1 , , , (6.15)Φ ( z ) = Φ ( ze πi ) ˆ S e πiθ ∞ σ , (6.16)where ˆ S k − = (cid:18) s k − (cid:19) , ˆ S k = (cid:18) s k (cid:19) , k = 1 , . (6.17)The solution Φ ( z, x ) has the asymptotic behaviors as z → ( z, x ) = Φ (0)1 ( x )( I + O ( z )) z θ σ ˆ E, with some invertible constant matrix ˆ E . The monodromy data of the system (6.10) is definedas the set n e πiθ ∞ , e πiθ ; ˆ S , ˆ S , ˆ S , ˆ S ; ˆ E o . (6.18)They satisfy the cyclic conditionˆ S ˆ S ˆ S ˆ S e πiθ ∞ σ = ˆ E − e − πiθ σ ˆ E. (6.19)Thus, the Stokes multipliers ˆ s k , k = 1 , , , e πiθ ∞ (1 + ˆ s ˆ s ) + e − πiθ ∞ (ˆ s ˆ s + (1 + ˆ s ˆ s )(1 + ˆ s ˆ s )) = 2 cos(2 πθ ) . (6.20)Moreover, the cyclic condition (6.18) specifies the connection matrices ˆ E to within a left-multiplicative diagonal matrices; see [12]. 21 .3 Reduction of the linear system for PIV to BHE In this subsection, we will derive BHE from (1.11) associated with the isomonodromy systemof PIV when the PIV transcendents y ( x ) tends to zero or infinity.Substituting A ( z, x ) into (1.11) givesd Φ d z − (cid:18) z − λ − z (cid:19) dΦ d z − Q ( z, x )Φ = 0 , (6.21)where Q ( z, x ) = z + 2 xz + x + 2(1 − θ ∞ ) + θ z − z − λ (cid:18) z + x + θ − vz (cid:19) + 1 z (cid:20) y v − (cid:18) y + 2 x + 4 θ y (cid:19) v + ( θ + θ ∞ ) y + (1 + 2 θ ) x (cid:21) = z + 2 xz + x + 2(1 − θ ∞ ) + θ z − z − λ (cid:18) z + x + θ − vz (cid:19) + 1 z (cid:20) dd x log τ ( x ) + (1 − θ ∞ ) x (cid:21) . To derive BHE, we need to eliminate the extra singularity λ = y/
2. Thus we consider thezeros and poles of y . According to [23, (1.4)-(1.5)], the solution y of PIV equation admits theTaylor expansion near a zero y ( x ) = ε ( x − a ) + b ( x − a ) + O (( x − a ) ) , (6.22)the Laurent expansion near a pole y ( x ) = σ ( x − a ) − − a + a ( x − a ) + b ( x − a ) + O (( x − a ) ) , (6.23)where b is arbitrary and the parameters ε = ε ± = ± θ , σ = σ ± = ± , a = 13 σ (cid:0) a − θ ∞ − σ ) (cid:1) . Let us first consider the zeros of y . It follows from (6.11) and (6.22) that v ( x ) = ( O ( x − a ) , ε = ε + , θ + O ( x − a ) , ε = ε − . For ε = ε + , taking x → a in (6.21) and then settingΦ = z θ e z + az w, we obtain the BHE d w d z + (cid:18) z + 2 a + 2 θ z (cid:19) d w d z + pz − qz w = 0 ,p = 2( θ + θ ∞ ) ,q = lim x → a dd x log τ ( x ) − θ + θ ∞ ) a. (6.24)For ε = ε − , by taking x → a in (6.21) and settingΦ = z θ +1 e z + az w,
22e obtain the BHE d w d z + (cid:18) z + 2 a + 2 θ + 2 z (cid:19) d w d z + pz − qz w = 0 ,p = 2( θ + θ ∞ + 1) ,q = lim x → a dd x log τ ( x ) − θ + θ ∞ + 1) a. (6.25)Next we focus on the poles of y . It is seen from (6.11) and (6.23) that v ( x ) = ( ( x − a ) − + O (1) , σ = σ + ,θ + θ ∞ + O ( x − a ) , σ = σ − . (6.26)By taking the limit x → a in (6.21) and then making the transformationΦ = z θ e z + az w, we obtain another two BHEs d w d z + (cid:18) z + 2 a + 2 θ + 1 z (cid:19) d w d z + pz − qz w = 0 ,p = 2( θ + θ ∞ + 1) ,q = lim x → a (cid:18) dd x log τ ( x ) − x − a (cid:19) − (2 θ + 2 θ ∞ + 1) a = − b − (cid:18) θ + 2 θ ∞ − (cid:19) a (6.27)for σ = σ + = 1, d w d z + (cid:18) z + 2 a + 2 θ + 1 z (cid:19) d w d z + pz − qz w = 0 ,p = 2( θ + θ ∞ ) ,q = lim x → a dd x log τ ( x ) − θ + θ ∞ ) a = − b − (cid:18) θ + 2 θ ∞ + 12 (cid:19) a (6.28)for σ = σ − = − We have shown that the BHE (1.4), with the accessory parameters a and q , the other fixedparameters γ and p can be obtained as a limit of the first row of the isomonodromy familyof Φ( z, x ), when x → a , a being one of the poles or zeros of the Painlev´e IV transcendents.While the accessory parameters ( a, q ) are expressed in terms of the parameters in the Laurentor Taylor expansion of the PIV transcendents as given in (6.24)-(6.28). Therefore, we have thefollowing description of the isomonodromy set of accessory parameters of BHE. Theorem 5.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that theBHE (1.4) corresponding to these parameters has the same monodromy data described by (6.9) as the original equation with the parameters a and q . Under the bijection given in the lastequation of (6.27) , this set coincides with the set of parameters ( a n , b n ) in the Laurent expansionnear the poles a n of the unique solution of PIV (6.12) with the parameter θ = ( γ − / , θ ∞ = ( p − γ − / and the same monodromy data (6.9) as the BHE. Accessory parameters of CHE
Consider the CHE (1.2) with the accessory parameters a and q , the fixed parameters γ , δ and p . There exist uniquely two linear independent solutions of (1.2), namely Y k = ( y k , y k ),normalizing the asymptotic behavior as z → ∞ Y k ( z ) ∼ e − z z − ( θ + θ ) (1 , /z ) e z σ z − θ ∞ σ , z ∈ Ω k , (7.1)where the parameters θ = γ , θ = δ − θ ∞ = 2 p − γ − δ + 1 and the Stokes sectors Ω k , k = 1 , k = n z ∈ C : π − k + 1) < arg z < π − k + 5) o . (7.2)The solutions are related to each other by the Stokes matrices Y ( z ) = Y ( z ) S , S = (cid:18) s (cid:19) , (7.3) Y ( ze πi ) e πi ( θ + θ ) e πiθ ∞ σ = Y ( z ) S , S = (cid:18) s (cid:19) . (7.4)Near the regular singular points, Y has the asymptotic behaviors Y ( z ) ∼ (cid:16) z , z − θ (cid:17) E , Y ( z ) ∼ (cid:16) ( z − a ) , ( z − a ) − θ (cid:17) E , with some invertible constant matrices E and E . After an analytic continuation along a closedloop around a singular point, we obtain the relations Y (cid:0) z k + e πi ( z − z k ) (cid:1) = Y ( z ) e − πiθ k M k , M k = E − k e πiθ k σ E k (7.5)for k = 0 ,
1, where z = 0 and z = a . We define the monodromy data of the CHE equation as n e πiθ ∞ , e πiθ , e πiθ ; S , S ; E , E o . (7.6)According to (7.3)-(7.5), the monodromy matrices satisfies the cyclic condition M M = S S e − πiθ ∞ σ . (7.7)Suppose 2 cos( πσ ) = Tr M M = 2 cos( πθ ∞ ) + s s e πiθ ∞ , (7.8)with 0 ≤ Re σ ≤
1. Then, the connection matrices can be parameterized in terms of σ and theother free parameter s ; see [19]. For instance, we have for σ = 0 [19, Equation (3.7)]: D E D = (cid:18) sin( π ( θ + θ + σ )) sin( π ( θ − θ + σ )) − s sin( π ( θ + θ − σ )) sin( π ( θ − θ − σ )) − s − (cid:19) × e − πiσ sin( π ( θ ∞ + σ )) e πiσ sin( π ( θ ∞ − σ )) ! , (7.9)and D E D = (cid:18) sin( π ( θ + θ + σ )) − se − πiσ sin( π ( θ + θ − σ )) − s − e πiσ sin( π ( θ − θ + σ )) sin( π ( θ − θ − σ )) (cid:19) × e − πiσ sin( π ( θ ∞ + σ )) e πiσ sin( π ( θ ∞ − σ )) ! , (7.10)with some invertible diagonal matrices D , D and D .24 .2 Isomonodromy deformation and PV equation To study the isomonodromy deformation of the CHE, it is convenient to consider a 2 × ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) , A ( z, x ) = σ A ( x ) z + A ( x ) z − x ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , B ( z, x ) = − A ( x ) z − x , (7.11)the coefficients A ( x ) = (cid:18) v + θ − u ( v + θ ) vu − v − θ (cid:19) ,A ( x ) = − v − θ + θ ∞ yu (cid:0) v + θ − θ + θ ∞ (cid:1) − yu (cid:0) v + θ + θ + θ ∞ (cid:1) v + θ + θ ∞ ! . The compatibility condition of the above Lax pair reads x d y d x = xy − v ( y − − ( y − (cid:18) θ − θ + θ ∞ y − θ + θ + θ ∞ (cid:19) ,x d v d x = yv (cid:18) v + θ − θ + θ ∞ (cid:19) − y ( v + θ ) (cid:18) v + θ + θ + θ ∞ (cid:19) ,x d u d x = u (cid:20) − v − θ − θ ∞ + y (cid:18) v + θ − θ + θ ∞ (cid:19) + 1 y (cid:18) v + θ + θ + θ ∞ (cid:19)(cid:21) , (7.12)which implies the PV equationd y d x = (cid:18) y + 1 y − (cid:19) (cid:18) d y d x (cid:19) − x d y d x + ( y − x (cid:18) αy + βy (cid:19) + γyx − y ( y + 1)2( y − , (7.13)where α = 18 ( θ − θ + θ ∞ ) , β = −
18 ( θ − θ − θ ∞ ) , γ = 1 − θ − θ . There exist unique solutions Φ k ( z, x ) of the equation (7.11), which satisfies the normalizedasymptotic behavior as z → ∞ Φ k ( z, x ) = ( I + O (1 /z )) e z σ z − θ ∞ σ , z ∈ Ω k , (7.14)where Ω k , k = 1 ,
2, are the Stokes sectors defined in (7.2). The solutions are related to eachother by the Stokes matrices Φ ( z ) = Φ ( z ) ˆ S , ˆ S = (cid:18) s (cid:19) , (7.15)Φ ( ze πi ) e πiθ ∞ σ = Φ ( z ) ˆ S , ˆ S = (cid:18) s (cid:19) . (7.16)Near the regular singular points z = 0 and z = a , Φ ( z, x ) has the asymptotic behaviorsΦ ( z, x ) = Φ ( k )1 ( x ) ( I + O ( z − z k )) ( z − z k ) θ k σ ˆ E k , z → z k , E k and z -independent matrices Φ ( k )1 ( x ), k = 0 ,
1. Themonodromy data of the system (7.11) is defined as the set n e πiθ ∞ , e πiθ k ; ˆ S , ˆ S ; ˆ E , ˆ E o . (7.17)They satisfy the cyclic conditionˆ M ˆ M = ˆ S ˆ S e − πiθ ∞ σ , ˆ M k = ˆ E − k e πiθ k σ ˆ E k , k = 0 , . (7.18) In this subsection, we will derive CHE from the linear systems (1.11) and (1.12) for PV at thepoles, zeros and 1-points (that is, y ( a ) = 1) of the solutions to PV.Substituting the elements of A ( z, x ) into (1.11) givesd Φ d z − (cid:18) z − λ − z − z − x (cid:19) dΦ d z − P ( z, x )Φ = 0 , (7.19)where λ = λ ( x ) = x − y ( v + ( θ − θ + θ ∞ )) v + θ ,P ( z, x ) = 14 + θ z + M ( x ) z ( z − x ) + θ z − x ) + 2 − θ ∞ z − x ) − z − λ (cid:18)
12 + 2 v + θ z − v + θ + θ ∞ z − x ) (cid:19) ,M ( x ) = yv (cid:18) v + θ − θ + θ ∞ (cid:19) + 1 y ( v + θ ) (cid:18) v + θ + θ + θ ∞ (cid:19) − (cid:18) v + θ (cid:19) (cid:18) v + θ + θ ∞ (cid:19) − (cid:18) v + 1 + θ (cid:19) x − θ ∞ . In virtue of the definition of the PV τ -function (see [20, (C.43)]): x dd x log τ ( x ) = − (cid:20) v − y (cid:18) v + θ + θ + θ ∞ (cid:19)(cid:21) (cid:20) v + θ − y (cid:18) v + θ − θ + θ ∞ (cid:19)(cid:21) − (cid:18) v + θ + θ ∞ (cid:19) x, (7.20)the expression of M ( x ) is simplified to M ( x ) = x dd x log τ ( x ) + ( θ ∞ − x − θ + θ − θ ∞ − θ ∞ . Similarly, substituting the elements of A ( z, x ) into (1.12) givesd Φ d z − (cid:18) z − λ − z − z − x (cid:19) dΦ d z − Q ( z, x )Φ = 0 , (7.21)where λ = λ ( x ) = x − v + ( θ + θ + θ ∞ ) yv ,Q ( z, x ) = 14 + θ z + N ( x ) z ( z − x ) + θ z − x ) − θ ∞ z − x )+ 1 z − λ (cid:18)
12 + 2 v + θ z − v + θ + θ ∞ z − x ) (cid:19) ,N ( x ) = x dd x log τ ( x ) + ( θ ∞ + 1) x − θ + θ − θ ∞ θ ∞ .
26o eliminate the singularities z = λ and z = λ and obtain CHE, we need λ ( x ) = 0 , or λ ( x ) = x, or λ ( x ) = ∞ , (7.22) λ ( x ) = 0 , or λ ( x ) = x, or λ ( x ) = ∞ , (7.23)in (7.19) and (7.21), respectively. For this purpose, we consider the poles, zeros and 1-points ofthe solution y .According to [16, (36.4)-(36.7)], the solution y admits the Laurent expansion near a pole y ( x ) = ( ε ( x − a ) − + b + O ( x − a ) , α = 0 ,ab ( x − a ) − + b ( x − a ) − + O (1) , α = 0 , and the following Taylor expansions near a zero and near a 1-point y ( x ) = ( δ ( x − a ) + b ( x − a ) + O (( x − a ) ) , β = 0 ,b ( x − a ) + O (( x − a ) ) , β = 0 ,y ( x ) = 1 + ω ( x − a ) + (cid:18)
12 + ω − θ + θ a (cid:19) ( x − a ) + O (( x − a ) ) , where b is arbitrary and the parameters ε = ε ± = ± a/ ( θ − θ + θ ∞ ) ,δ = δ ± = ± ( θ − θ − θ ∞ ) / a,ω = ω ± = ± . (7.24)From the first equation in the compatibility condition (7.12) and above expansions, we derivethe behaviors of v, λ , λ at each pole, zero or 1-point of y , as given in Table 1.Case y v λ λ ε = ε + simple pole 0 0 O ( a ) ε = ε − simple pole − ( θ − θ + θ ∞ ) O ( a ) aα = 0 double pole 0 0 aδ = δ + simple zero − θ O ( a ) 0 δ = δ − simple zero − ( θ + θ + θ ∞ ) a O ( a ) β = 0 double zero − ( θ + θ + θ ∞ ) a ω = ω + O (1) O ( a ) O ( a ) ω = ω − a ( x − a ) − + O (1) ∞ ∞ Table 1: Behaviors of y, v, λ , λ at the critical point x = a It is readily seen from Table 1 that the condition (7.22) or (7.23) is fulfilled when a is apole, zero or 1-point of y with y ′ ( a ) = ω − . Thus, the equation (7.19) or (7.21) is equivalent toCHE in these cases. For example, we consider the case ε = ε + . According to Table 1, we have v ( a ) = 0, λ ( a ) = 0. Thus, by taking x → a in (7.19) and settingΦ = z θ ( z − a ) θ e z w,
27e obtain the CHE d w d z + (cid:18) θ z + 1 + θ z − a (cid:19) d w d z + pz − qz ( z − a ) w = 0 ,p = 12 ( θ + θ + θ ∞ ) ,q = lim x → a x dd x log τ ( x ) + ( θ + θ ∞ ) a − ( θ + θ ) − θ ∞ a θ − θ + θ ∞ ) − b θ − θ + θ ∞ ) + 14 ( θ ∞ + 1)( θ − θ + θ ∞ ) − θ θ . (7.25)Actually, we obtain six different CHEs with the characteristic exponents at the singularpoints shown in Table 2 and the accessory parameters given below: q = lim x → a x dd x log τ ( x ) + ( θ + θ ∞ ) a − ( θ + θ ) − θ ∞ , (7.26)for ε = ε + , δ = δ + and δ = δ − , q = lim x → a x dd x log τ ( x ) + ( θ + θ ∞ + 2) a − ( θ + θ ) − θ ∞ , (7.27)for ε = ε − , q = lim x → a (cid:18) x dd x log τ ( x ) − ax − a (cid:19) + ( θ + θ ∞ ) a − ( θ + θ ) − θ ∞ − θ − θ , (7.28)for ω = ω − and employing Φ , q = lim x → a (cid:18) x dd x log τ ( x ) − ax − a (cid:19) + ( θ + θ ∞ + 2) a − ( θ + θ ) − θ ∞ − θ − θ , (7.29)for ω = ω − and using Φ .Case Equation 0 a ∞ pε = ε + (7.19) θ θ ( θ + θ + θ ∞ ) ε = ε − (7.21) 1 + θ θ ( θ + θ + θ ∞ + 2) δ = δ + (7.21) θ θ ( θ + θ + θ ∞ + 2) δ = δ − (7.19) 1 + θ θ ( θ + θ + θ ∞ ) ω = ω − (7.19) 1 + θ θ ( θ + θ + θ ∞ + 2) ω = ω − (7.21) 1 + θ θ ( θ + θ + θ ∞ + 2)Table 2: The characteristic exponents of CHEs obtained from equations (7.19) and (7.21)We mention that the case λ ( x ) = x has been considered in [6, 7]. According to Table 1,the condition λ ( x ) = x is equivalent to the case that y has simple zeros with coefficient δ = δ − or y has double zeros. We have shown that the CHE can be obtained as a limit of the first row or second row of theisomonodromy family of Φ( z, x ) when x → a , a being one of the poles, zeros or 1-points of28he Painlev´e V transcendents corresponding to the same monodromy data of CHE. While theaccessory parameters ( a, q ) are expressed in terms of the parameters in the Laurent or Taylorexpansion of the PV transcendents. As a consequence, we have the following description of theisomonodromy set of accessory parameters of CHE. Theorem 6.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that theCHE (1.2) corresponding to these parameters and the fixed parameters γ , δ and p , has the samemonodromy data (7.6) as the original one with the parameters a and q . Under the relationgiven in the last equation of (7.25) , this set coincides with the set of parameters ( a n , b n ) in theLaurent expansion near the poles a n of the unique solution of PV (7.13) with the parameters θ = γ , θ = δ − , θ ∞ = 2 p − γ − δ + 1 and corresponding to the same monodromy data (7.6) as the CHE. In this subsection, we will derive HE from the linear system (1.11) for PVI at the poles, zeros,1-points ( y ( x ) = 1) and fixed points ( y ( x ) = x ) of the solutions of PVI.Consider the following Lax pair for PVI equation (see [20, (C.46)-(C.47)]): ∂ Φ( z, x ) ∂z = A ( z, x )Φ( z, x ) , A ( z, x ) = A z + A z − A z − x ,∂ Φ( z, x ) ∂x = B ( z, x )Φ( z, x ) , B ( z, x ) = − A z − x , (8.1)where A i = v i + θ i − u i v i u − i ( v i + θ i ) − v i ! , i = 0 , , . If we set κ = −
12 ( θ + θ + θ − θ ∞ ) ,κ = −
12 ( θ + θ + θ + θ ∞ ) ,A + A + A = − κ κ ! ,A ( z, x ) = − u v z + − u v z − − u v z − x = k ( z − y ) z ( z − z − x ) ,v = v + θ y + v + θ y − v + θ y − x , (8.2)then the compatibility condition of the above Lax pair gives d y d x = y ( y − y − x ) x ( x − (cid:18) v − θ y − θ y − − θ − y − x (cid:19) , d v d x = 1 x ( x − (cid:26) (cid:2) − y + 2(1 + x ) y − x (cid:3) v + (cid:2) (2 y − − x ) θ + (2 y − x ) θ + (2 y −
1) ( θ − (cid:3) v − κ ( κ + 1) (cid:27) , d k d x = k ( θ ∞ − y − xx ( x − , (8.3)29hich leads to the PVI equationd y d x = 12 (cid:18) y + 1 y − y − x (cid:19) (cid:18) d y d x (cid:19) − (cid:18) x + 1 x − y − x (cid:19) d y d x + y ( y − y − x ) x ( x − (cid:18) α + β xy + γ x − y − + δ x ( x − y − x ) (cid:19) , (8.4)where α = 12 ( θ ∞ − , β = − θ , γ = 12 θ , δ = 12 (cid:0) − θ (cid:1) . The PVI τ -function is defined by (see [20, (C.57)]): x ( x −
1) dd x log τ ( x ) = y ( y − y − x ) (cid:26) v − (cid:18) θ y + θ y − θ y − x (cid:19) v + κ κ y ( y − (cid:27) + θ θ ( x −
1) + θ θ x. (8.5)Substituting the entries of A ( z, x ) into (1.11) givesd Φ d z + P ( z, x ) dΦ d z + Q ( z, x )Φ = 0 , (8.6)where P ( z, x ) = 1 − θ z + 1 − θ z − − θ z − x − z − y ,Q ( z, x ) = κ ( κ + 1) z ( z − − M ( x ) z ( z − z − x ) + y ( y − vz ( z − z − y ) ,M ( x ) = x ( x −
1) dd x log τ ( x ) + κ ( y − x ) + y ( y − v − θ θ ( x − − θ θ x. It is clear that the extra singularity z = y can be removed by considering the critical values y ( a ) = 0 , or y ( a ) = 1 , or y ( a ) = a, or y ( a ) = ∞ . First, we consider the critical value y ( a ) = a . By taking the limit x → a in (8.6), we obtainthe following HE directly d Φ d z + (cid:18) − θ z + 1 − θ z − − θ z − a (cid:19) dΦ d z + pz − qz ( z − z − a ) Φ = 0 ,p = 14 ( θ + θ + θ − θ ∞ )( θ + θ + θ + θ ∞ − ,q = lim x → a x ( x −
1) dd x log τ ( x ) + κ ( κ + 1) a − θ θ ( a − − θ θ a. (8.7)The equation (8.7) has also been obtained in [1, 3, 8]. It was also noted therein that thecondition y ( a ) = a can be expressed in terms of the associated τ -function.Next, we consider the poles of y . Let x = a ( a = 0 ,
1) be a movable pole of y ( x ). Then y ( x )possesses the following Laurent expansions (see [16, (46.7)]): y ( x ) = εx − a + b + O ( x − a ) , α = 0 ,b ( x − a ) + (2 a − ba ( a − x − a ) + c + O ( x − a ) , α = 0 , (8.8)where ε = ε ± = ± a ( a − / ( θ ∞ − b is arbitrary and c = 13 ( a + 1) + b a ( a − n a ( a −
1) + 1 − aθ + ( a − θ − a ( a − θ − o . We derive HE by considering each case separately.30 ase 1: a is a simple pole with residue ε = ε + and θ ∞ = 0 In this case, we have from (8.5), (8.6) and (8.8) that M ( x ) = x ( x −
1) dd x log τ ( x ) + ( θ ∞ − b − κ a − θ θ ( a − − θ θ a + 12 (cid:2) − θ + ( θ − a − − a + 1 − ( θ ∞ − a (cid:3) + O ( x − a ) , (8.9)and x ( x −
1) dd x log τ ( x ) = θ ∞ (1 − θ ∞ ) b − θ ∞ (cid:2) − θ + ( θ − a − − a + 1 − ( θ ∞ − a (cid:3) − κ ( a − − κ κ a − κ (cid:2) aθ + ( a − θ (cid:3) + θ θ ( a −
1) + θ θ a + O ( x − a ) . (8.10)This, together with θ ∞ = 0, implies that M ( x ) = (1 − θ − ∞ ) x ( x −
1) dd x log τ ( x ) − κ a − θ − ∞ n κ ( a −
1) + κ κ a + κ (cid:2) aθ + ( a − θ (cid:3)o − (1 − θ − ∞ ) (cid:2) θ θ ( a −
1) + θ θ a (cid:3) + O ( x − a ) . Taking x → a in (8.6), we obtain the HE d Φ d z + (cid:18) − θ z + 1 − θ z − − θ z − a (cid:19) dΦ d z + pz − qz ( z − z − a ) Φ = 0 ,p = 14 ( θ + θ + θ − θ ∞ )( θ + θ + θ + θ ∞ − ,q = lim x → a (1 − θ − ∞ ) x ( x −
1) dd x log τ ( x ) + κ ( κ + 2) a − θ − ∞ n κ ( a − κ κ a + κ (cid:2) aθ + ( a − θ (cid:3)o − (1 − θ − ∞ ) (cid:2) θ θ ( a −
1) + θ θ a (cid:3) , (8.11)where κ and κ are given in (8.2). Case 2: a is a simple pole with residue ε = ε + and θ ∞ = 0 When θ ∞ = 0, it is seen from (8.9) and (8.10) that M ( x ) = x ( x −
1) dd x log τ ( x ) − b + 12 (cid:2) − θ + ( θ − a − − a (cid:3) − κ a − θ θ ( a − − θ θ a + O ( x − a ) , and x ( x −
1) dd x log τ ( x ) = − κ ( a − − κ κ a − κ (cid:2) aθ + ( a − θ (cid:3) + θ θ ( a −
1) + θ θ a + O ( x − a ) . It is straightforward that the free parameter b is independent of τ ( x ). To express the accessoryparameter in terms of the τ -function as (8.11), we need the Schlesinger transformation used by31ubrovin and Kapaev to shift the formal monodromy at infinity θ ∞ by −
2, i.e. θ ∞ θ ∞ − d Φ d z + (cid:18) − θ z + 1 − θ z − − θ z − a (cid:19) dΦ d z + pz − qz ( z − z − a ) Φ = 0 ,p = 14 ( θ + θ + θ + 2)( θ + θ + θ − ,q = lim x → a x ( x −
1) dd x log ˜ τ ( x ) + ˜ κ (˜ κ + 2) a + 12 n ˜ κ ( a − κ ˜ κ a + ˜ κ (cid:2) aθ + ( a − θ (cid:3)o − (cid:2) θ θ ( a −
1) + θ θ a (cid:3) , where ˜ τ ( x ) is a new τ -function defined by x ( x −
1) dd x log ˜ τ ( x ) = y ( y − y − x ) (cid:26) v − (cid:18) θ y + θ y − θ y − x (cid:19) v + ˜ κ ˜ κ y ( y − (cid:27) + θ θ ( x −
1) + θ θ x with ˜ κ = −
12 ( θ + θ + θ + 2) , ˜ κ = −
12 ( θ + θ + θ − . Case 3: a is a simple pole with residue ε = ε − From (8.6) and (8.8), it follows that M ( x ) = x ( x −
1) dd x log τ ( x ) − a ( a − x − a + 12 ( θ + θ + 2 θ − a + 1 −
12 ( θ + θ ) − θ θ ( a − − θ θ a + O ( x − a ) . By taking x → a in (8.6), we arrive at the HE d Φ d z + (cid:18) − θ z + 1 − θ z − − θ z − a (cid:19) dΦ d z + pz − qz ( z − z − a ) Φ = 0 ,p = 14 ( θ + θ + θ − θ ∞ − θ + θ + θ + θ ∞ − ,q = lim x → a (cid:18) x ( x −
1) dd x log τ ( x ) − a ( a − x − a (cid:19) + ( κ + 1)( κ + 1) a + 12 ( θ + θ + 2 θ − a + 1 −
12 ( θ + θ ) − θ θ ( a − − θ θ a = − (1 − θ ∞ ) b − (cid:16) θ + θ ) − ( θ + θ ) + θ − ( θ ∞ − + 4( θ ∞ − (cid:17) a −
12 ( θ + θ ) + θ ∞ − (cid:16) ( θ + θ ) − θ + ( θ ∞ − (cid:17) , (8.12)where κ and κ are given in (8.2). 32 ase 4: a is a double pole We see from (8.6) and (8.8) that M ( x ) = x ( x −
1) dd x log τ ( x ) − a ( a − x − a −
12 ( θ + θ − a −
12 ( θ + θ ) − θ θ ( a − − θ θ a + O ( x − a ) . By taking x → a in (8.6), we arrive at the HE d Φ d z + (cid:18) − θ z + 1 − θ z − − θ z − a (cid:19) dΦ d z + pz − qz ( z − z − a ) Φ = 0 ,p = 14 ( θ + θ + θ − θ + θ + θ − ,q = lim x → a (cid:18) x ( x −
1) dd x log τ ( x ) − a ( a − x − a (cid:19) + κ ( κ + 2) a −
12 ( θ + θ − a −
12 ( θ + θ ) − θ θ ( a − − θ θ a = a ( a − b + (cid:16) κ ( κ + θ + 3) + 3 (cid:17) a + (cid:16) κ ( κ + θ + 1) − (cid:17) , (8.13)where κ and κ are given in (8.2), with θ ∞ = 1 in this case.It is noted that the above HEs are also obtained from the linear system at the poles ofthe solutions of PVI in [10], while the accessory parameters in these equations are expressed interms of the free parameter of the Laurent expansion of y at the poles.Finally, we consider the critical values y ( x ) = 0 and y ( x ) = 1. It follows from [16, (46.8)-(46.9)] that y admits the following Taylor expansions: y ( x ) = ( λ ( x − a ) + b ( x − a ) + O (( x − a ) ) , β = 0 ,b ( x − a ) + O (( x − a ) ) , β = 0 , (8.14) y ( x ) = ( ω ( x − a ) + b ( x − a ) + O (( x − a ) ) , γ = 0 , b ( x − a ) + O (( x − a ) ) , γ = 0 , (8.15)where b is arbitrary and λ = λ ± = ± θ a − ,ω = ω ± = ± θ a . It is seen from (8.6), (8.14) and (8.15) that M ( x ) = x ( x −
1) dd x log τ ( x ) − θ − κ a − θ θ ( a − − θ θ a + O ( x − a ) , for λ = λ + , M ( x ) = x ( x −
1) dd x log τ ( x ) − κ a − θ θ ( a − − θ θ a + O ( x − a ) , for λ = λ − , M ( x ) = x ( x −
1) dd x log τ ( x ) − θ − κ a − θ θ ( a − − θ θ a + O ( x − a ) , β = 0, M ( x ) = x ( x −
1) dd x log τ ( x ) + κ (1 − a ) − θ θ ( a − − θ θ a + O ( x − a ) , for ω = ω + , M ( x ) = x ( x −
1) dd x log τ ( x ) + θ + κ (1 − a ) − θ θ ( a − − θ θ a + O ( x − a ) , for ω = ω − , M ( x ) = x ( x −
1) dd x log τ ( x ) + θ κ (1 − a ) − θ θ ( a − − θ θ a + O ( x − a ) , for γ = 0.Taking x → a in (8.6), we then obtain six HEs. We show the exponent parameters at thesingularities 0, 1, a and ∞ in Table 3 and list the accessory parameters below: q = lim x → a x ( x −
1) dd x log τ ( x ) + θ ( θ −
1) + ( κ κ + θ − θ θ − θ θ − θ θ ) a, for λ = λ + , q = lim x → a x ( x −
1) dd x log τ ( x ) + θ θ + ( κ κ − θ θ − θ θ ) a, for λ = λ − , q = lim x → a x ( x −
1) dd x log τ ( x ) + ( κ κ − θ θ ) a, for β = 0, q = lim x → a x ( x −
1) dd x log τ ( x ) + κ + θ θ + ( κ κ − θ θ − θ θ ) a, for ω = ω + , q = lim x → a x ( x −
1) dd x log τ ( x ) + θ + θ θ + κ + ( κ κ + θ − θ θ − θ θ − θ θ ) a, for ω = ω − , q = lim x → a x ( x −
1) dd x log τ ( x ) + κ + θ θ + ( κ κ − θ θ ) a, for γ = 0.Case 0 1 a ∞ λ = λ + θ − θ − θ ( θ − θ − θ + θ ∞ )( θ − θ − θ − θ ∞ + 2) λ = λ − − θ − θ − θ ( θ + θ + θ − θ ∞ )( θ + θ + θ + θ ∞ − β = 0 0 1 − θ − θ ( θ + θ − θ ∞ )( θ + θ + θ ∞ − ω = ω + − θ − θ − θ ( θ + θ + θ − θ ∞ )( θ + θ + θ + θ ∞ − ω = ω − − θ θ − θ ( θ − θ − θ + θ ∞ )( θ − θ − θ − θ ∞ + 2) γ = 0 1 − θ − θ ( θ + θ − θ ∞ )( θ + θ + θ ∞ − λ = λ + and ω = ω − , use has be made of the transformationsΦ = z θ w and Φ = ( z − θ w to get the canonical form of HE (1.1).34 .2 Isomonodromy set of accessory parameters of HE Consider the HE (1.1) with parameters a and q . There exist uniquely two linear independentsolutions of (1.1), namely ( y , y ), satisfied the normalized asymptotic behavior as z → ∞ :( y ( z ) , y ( z )) ∼ (cid:16) (1 /z ) α , (1 /z ) β (cid:17) , (8.16)where α and β are the characteristic exponents at z = ∞ . We have the asymptotic behaviorsof ( y , y ) near the singular points z = 0 , z = 1 , z = a ( y ( z ) , y ( z )) ∼ (cid:16) ( z − z k ) θ k , ( z − z k ) (cid:17) E k , k = 0 , , , with the characteristic exponents θ = 1 − γ , θ = 1 − δ , θ = 1 − ǫ and some invertibleconstant matrix E k , k = 0 , ,
2. Under an analytic continuation along a closed loop around asingular point, we obtain another two linear independent solutions of the same equation, whichare therefore related to ( y , y ) by (cid:0) y (cid:0) z k + e πi ( z − z k ) (cid:1) , y (cid:0) z k + e πi ( z − z k ) (cid:1)(cid:1) = ( y ( z ) , y ( z )) e πiθ k M k , (cid:0) y ( e πi z ) , y ( e πi z ) (cid:1) = ( y ( z ) , y ( z )) e − πi ( α + β ) M ∞ . Here the constant matrices are known as the monodromy matrices and determined by theconnection matrices and the characteristic exponents M k = E − k e πiθ k σ E k , M ∞ = e πiθ ∞ σ , with k = 0 , , θ ∞ = − α + β . The monodromy data of HE (1.1) is then constituted by n e πiθ ∞ , e πiθ , e πiθ , e πiθ ; E , E , E o , (8.17)where the characteristic exponents are related to the fixed parameters in (1.1) by θ = 1 − γ , θ = 1 − δ , θ = 1 − ǫ and θ ∞ = − α + β . The monodromy matrices satisfies the cyclic condition M ∞ M M M = I. According to [19], the monodromy matrices can be written explicitly in terms of the character-istic exponents and the parameters2 cos( πσ jk ) = Tr M j M k = Tr M k M j with j, k = 0 , , j < k . Moreover, by the cyclic condition, only two of the parameters { σ , σ , σ } are independent. Therefore, we get the same number of independent parametersof the monodromy matrices as the parameters a and q in HE equation.To study the isomonodromy deformation of the HE equation, it is convenient to considerthe matrix Fuchsian system with four regular singular points. As shown in Section 8.1, theisomonodromy deformation of (8.1) is described by the PVI equation (8.3). Moreover, let Φ( z, x )be the fundamental solution of (8.1) normalized at infinity, then Φ( z, x ) has the asymptoticbehaviors near the singular points:Φ( z, x ) = Φ ( k )0 ( x )( I + O ( z − z k ))( z − z k ) θ k ( z − z k ) θ k σ ˆ E k , and Φ( z, x ) = ( I + O (1 /z )) z α + β z − θ ∞ σ , z = 0, z = 1, z = x . Here the connection matrix ˆ E k are certain invertible constantmatrices. Similarly, the analytic continuation of Φ( z, x ) along a closed loop around the singularpoints are related to Φ( z, x ) by the monodromy matricesˆ M ∞ = e πiθ ∞ σ , ˆ M k = ˆ E − k e πiθ k σ ˆ E k , with k = 0 , ,
2. While the HE can be obtained from a family of isomonodromy deformationsystem by taking certain limit procedure at the poles of the solution of PVI equation; see forinstance (8.12). In this way, we obtain a family of accessory parameters sharing the samemonodromy data as stated in the following theorem.
Theorem 7.
There is a discrete set of pairs of accessory parameters ( a n , q n ) such that the HE (1.1) corresponding to these parameters has the same monodromy data (8.17) as the originalone with the parameters a and q . Under the bijection given in the last equation of (8.12) , andthat of (8.13) , this set coincides with the set of parameters ( a n , b n ) in the Laurent expansion (8.8) near the poles a n of the unique solution of PVI equation (8.4) with the same monodromydata (8.17) as the HE.Proof. Consider the HE (1.1) with given accessory parameters a and q . There exist uniquefundamental solutions ( y , y ) to this equation with the normalized behavior at infinity (8.16).The solutions are corresponding to the monodromy data (8.17) as mentioned at the beginningof this section. On the other side, there is a unique solution Φ( z, x ) normalized at infinity ofthe system (8.1) with the parameters related to the fixed parameters of HE (1.1) by θ = 1 − γ , θ = 1 − δ , θ = 1 − ǫ and θ ∞ = − α + β . Here the x -dependence of Φ( z, x ) is describedby the unique solution of the PVI equation with the Laurent expansion as given in (8.8) withthe parameter b given by the last equation of (8.12) and the parameter θ ∞ = 1 therein. When θ ∞ = 1, we consider the Laurent expansion with a double pole at a in (8.8). The solution Φ( z, x )is corresponding to the monodromy data of the form n e πiθ ∞ , e πiθ , e πiθ , e πiθ ; ˆ E , ˆ E , ˆ E o whichis independent of x .Using the limiting procedure shown in (8.12), we obtain the HE (1.1) with accessory param-eters a and q as limit of the first row of the isomonodromy family Φ( z ; x ) as x → a . It followsfrom the x -independent of the monodromy data that E k = ˆ E k , k = 0 , ,
2. Thus, we have shownthat any given accessory parameters ( a, q ) is related by the last equation of (8.12) to the poleparameters ( a, b ) of the unique solution of PVI corresponding to the same monodromy data(8.17) when the parameter θ ∞ = 1. In the case θ ∞ = 1, we consider (8.13) and similar analysisapplies. By the meromorphic property of the PVI solution, the poles of each PVI solution arediscrete and hence the set of pairs of accessory parameters of HE sharing the same monodromydata. We complete the proof of Theorem 7. In this section, we will derive some asymptotic approximations for the isomonodromy sets ofaccessory parameters of some Heun class equations expressed in terms of the monodromy data.The derivation are based on the connection between the accessory parameters and the theLaurent or Taylor coefficients for the corresponding Painlev´e functions obtained in the previoussections and the asymptotic expansions for the Painlev´e transcendents known in the literature.The equations of Heun class we considered in the section include the RBHE, CHE, and DHE.36 .1 Asymptotics of the accessory parameter of RBHE
Consider the RBHE (1.7) with the accessory parameters ( a n , q n ) such that the equation corre-sponding to these parameters has the monodromy data (4.9) specified by s − = − e − απi , s = e − βπi , s = − e απi , (9.1)with iβ ∈ R and α > − /
2. The corresponding Painlev´e XXXIV transcendents and their associ-ated Lax pair play important roles in random matrix theory when a Fisher-Hartwig singularitylocated at an interior point where the density of the equilibrium density vanishes quadratically[4], or at the right edge of the spectrum where typically the density vanishes like a squareroot [18, 34]. The asymptotics of the Painlev´e XXXIV transcendents and their associated τ -functions have been worked out in [9, 18, 34]. Using these asympotics and the relation betweenthe isomonodromy set of accessory parameters and the corresponding Painlev´e XXXIV tran-scendents given in Theorem 3, we derive the asymptotic behavior of the accessory parameters( a n , q n ) expressed in terms of the given monodromy data (9.1). Theorem 8.
Let ( a n , q n ) be the sequence of accessory parameters such that the RBHE (1.7) corresponding to these parameters has the same monodromy data (9.1) , then we have the asymp-totic approximations as n → ∞ | a n | = 2 nπ + 2 iβ ln 3 nπ + 4 iβ ln 2 − α − β ) + 12 (2 α + 1) π + O (cid:18) ln nn (cid:19) , (9.2) and q ( a n ) = 2 βi | a n | − (cid:0) α − α + 3 β (cid:1) a − n + O (cid:16) | a n | − (cid:17) . (9.3) Proof.
According to [34, Theorem 2] and [9, Proposition 3.4], we have the asymptotic behaviorsfor the Painlev´e XXXIV transcendents y ( x ) and the associated dd x log τ ( x ), corresponding tothe Stokes multipliers (9.1), as x → −∞ : y ( x ) = 2 | α − β | p | x | cos (cid:18) θ ( x )2 + arg Γ(1 + α − β ) − π (cid:19) × cos (cid:18) θ ( x )2 + arg Γ( α − β ) + π (cid:19) + O ( x − ) , (9.4)and dd x log τ ( x ) = 2 βi | x | − | α − β | x sin ( θ ( x ) + 2 arg Γ( α − β ) + arg( α − β ))+ α − β x + O ( | x | − ) , (9.5)where θ ( x ) = 43 | x | − iβ ln | x | − απ − iβ ln 2 , and β ∈ i R .It follows from (9.4) that y ( x ) admits a sequences of simple zeros lying on the negative realaxis and tending to infinity with the asymptotic approximation given in (9.2). Moreover, theleading coefficient of the Taylor expansion of y ( x ) near the zero a n is 2 α . The relation in thesecond equation of (4.20), together with (9.2) and (9.5), then implies (9.3). This completes theproof of Theorem 8. 37 .2 Asymptotics of the accessory parameter of CHE In this subsection, we consider the CHE (7.25) with the accessory parameters ( a n , q n ) such thatthe equation corresponding to these parameters has the monodromy data (7.6) parameterizedin terms of σ and s as given in (7.9)-(7.10). In the seminal work [19], Jimbo derived theasymptotic expansion for the PV tanscendents and the associated τ -function corresponding tothe monodromy data. Using these asympotics and the relation between the isomonodromy setof accessory parameters and the corresponding Painlev´e V transcendents given in Theorem 6,we derive the asymptotic behavior of the accessory parameters ( a n , q n ) expressed in terms ofthe given monodromy data. Theorem 9.
Let ( a n , q n ) be the sequence of accessory parameters such that the CHE (7.25) corresponding to these parameters has the same monodromy data (7.6) parameterized in termsof σ and s as given in (7.9) - (7.10) , then we have the asymptotic approximations as n → ∞ ln | a n | ∼ (Re σ ln | c | + Im σ arg c − πn | Im σ | ) / | σ | , (9.6) and q ( a n ) = σ − ( θ + θ ) − (cid:20) ( θ ∞ − θ − ( θ − )4( σ −
1) + 1 − θ − θ ∞ (cid:21) a n + O ( a n ) , (9.7) where c = ( σ + θ ∞ )( σ + θ + θ )Γ(1+ σ ) Γ( ( θ + θ − σ )+1)Γ( ( θ − θ − σ )+1)Γ( ( θ ∞ − σ )+1)( σ − θ ∞ )( σ − θ − θ )Γ(1 − σ ) Γ( ( θ + θ + σ )+1)Γ( ( θ − θ + σ )+1)Γ( ( θ ∞ + σ )+1) 1 s with Im σ = 0 and s = 0 .Proof. According to [19], we have the following asymptotic expansion for the τ -function of PVas x → τ ( x ) ∼ const. x ( σ − θ ∞ ) × ( − θ ∞ (cid:0) σ + θ − θ (cid:1) σ x − ( σ − θ ∞ ) h ( σ − θ ) − θ i σ (1 + σ ) x ( ρx σ )+ ( σ + θ ∞ ) h ( σ + θ ) − θ i σ (1 − σ ) x ( ρx σ ) − + ∞ X j =2 x j j X k = − j c jk x kσ ) , (9.8) ρ = Γ(1 − σ ) Γ( ( θ + θ + σ ) + 1)Γ( ( θ − θ + σ ) + 1)Γ( ( θ ∞ + σ ) + 1)Γ(1 + σ ) Γ( ( θ + θ − σ ) + 1)Γ( ( θ − θ − σ ) + 1)Γ( ( θ ∞ − σ ) + 1) s. (9.9)Here the parameters σ = 0 and s are the parameterization of the monodromy data of thecorresponding CHE as given in (7.8)-(7.10).As mentioned in [19, Remark 1], the asymptotics of the solution y of PV can also be obtainedfrom the asymptotic analysis carried out therein. The small- x asymptotic expansion for y wasalso derived in [27]. For Im σ = 0, we see from the asymptotics of the PV tanscendents y thatit admits a sequence of simple poles { a n } n ∈ N such that a σn ∼ c , c = ( σ + θ ∞ )( σ + θ + θ )( σ − θ ∞ )( σ − θ − θ ) ρ − , as a n → . (9.10)The sequence of poles are clustering at zero along the spiral described by | σ | ln | a n | − Re σ ln | c | − Im σ arg c ∼ − πn | Im σ | , (9.11)38nd | σ | arg a n − Re σ arg c + Im σ ln | c | ∼ πn Re σ ; (9.12)see for instance [27, Theorem 2.8]. It can be checked that the residues of y at a n equal ε + asgiven in (7.24).Thus, from (9.9) and (9.11) we have (9.6). Substituting (9.8), (9.10) into (7.26), we obtain(9.7). This completes the proof of Theorem 9.It should be mentioned that similar formulae for the asymptotic approximations of theaccessory parameters as a → τ -function of PV, with applications in black holes; see [6, (2.62a)-(2.62c)]. In general, theasymptotic expansions of the PV solutions y ( x ) and the associated τ -functions near infinity arerather complicated. There exists no general asymptotic expansions for y ( x ) or τ ( x ) as x → ∞ except along some special rays: arg x = 0 , π/ , π, π/
2; see [2, 21, 28]. In [2, 28], the asymptoticsfor y ( x ) and the logarithmic derivative of the τ -functions as x → i ∞ are established. From theasymptotic expansions, it is shown in [28] that under certain conditions the PV solutions admitsequences of poles and of zeros lying on the imaginary axis and tending to i ∞ . Combining theseresults with the expressions of the accessory parameters of CHE in terms of the τ -functiongiven in (7.26)-(7.27), the asymptotic approximations of the isomonodromic set of accessoryparameters ( a n , q n ) corresponding to some special monodromy data may also be obtained. In the pioneering work of McCoy, Tarcy and Wu [24], the asymptotics and the connectionformulae for one-parameter family of solutions to PIII were derived rigorously. These solutionshave important applications in the analysis of two-dimensional Ising model [33]. More precisely,they showed in [24] that there are one-parameter family of solutions of PIII with the asymptotics y ( x ; ν, λ ) ∼ − λ Γ (cid:18) ν + 12 (cid:19) − ν x − ν − e − x (cid:16) ∞ X j =1 c j x j (cid:17) , x → + ∞ , (9.13)where parameters θ , θ ∞ satisfy the relation θ = θ ∞ − ν. For | λ | < /π , the asymptotic behavior of y ( x ; ν, λ ) as x → + is described by y ( x ; ν, λ ) ∼ B (2 x ) σ with the parameters σ and B are given as explicit functions of λ and ν , which are now knownas the connection formulae. When λ > /π and the parameter ν = 0, the asymptotic behaviorof y ( x ) as x → + was also derived in [24] y ( x ; 0 , λ ) = x µ sin n µ ln x − iµ ) o + O ( x ) , x → + , (9.14)where λ = 1 π cosh( πµ ) , µ > . For λ < − /π , the asymptotics of y ( x ) as x → + follows from (9.14) and the symmetry relation[24, (4.127)] y ( x ; ν, λ ) = 1 y ( x ; ν, − λ ) . (9.15)39he asymptotic formula for general parameter ν ∈ R was worked out in [11]. When λ > /π and ν = 0, it is readily seen from (9.14) that y ( x ; 0 , λ ) admits a sequence of zeros { c n } n ∈ N lyingon the positive real axis with x = 0 being a limiting point: c n ∼ (cid:26) − nπµ + arg Γ( iµ ) µ (cid:27) → + , n → ∞ . (9.16)Moreover, it is straightforward to check that the Taylor expansions of y at the zeros { c n } n ∈ N and { c n +1 } n ∈ N are corresponding to σ = σ + and σ = σ − in (5.29), respectively. It is also seenfrom the relation (9.15) that there are infinitely many poles of y ( x ; 0 , λ ) clustering at x = 0 onthe positive real axis when λ < − /π .Let us consider the DHE equation (1.3) with the accessory parameters a and q and the fixedparameters γ = 1 + θ and p = ( θ ∞ + θ ). Applying Theorem 4, there is an isomonodromy setof pairs of accessory parameters ( a n , q n ) such that the DHE equation corresponding to these pa-rameters have the same monodromy data as the Painlev´e III transcendent y ( x ; 0 , λ ) determinedby the asymptotic behavior (9.14). Moreover, the parameter a n = c n and the accessory param-eters q n are expressed in terms of the Laurent parameters of the PIII transcendents as given in(5.31). Combining (5.16), (5.31), (9.14) and (9.16), we obtain the asymptotic approximation ofthe accessory parameters as stated in the following theorem. Theorem 10.
Let ( a n , q n ) be the sequence of accessory parameters such that the DHE equation (1.3) corresponding to these parameters has the same monodromy data as the Painlev´e III tran-scendents y ( x ; 0 , λ ) determined by the asymptotic behavior (9.14) , then we have the asymptoticapproximations as n → ∞ a n ∼ (cid:26) − nπµ + arg Γ( iµ ) µ (cid:27) → + , (9.17) and q ( a n ) = − µ + 14 + O ( a n ) . (9.18) Here the parameter λ = π cosh( πµ ) and µ > .Remark . In Theorem 10, we have derived the asymptotics of a isomonodromy sequence ofaccessory parameters for DHE (1.3). The asymptotics are expressed in terms of the parametersin the behavior (9.14) of the corresponding Painlev´e III transcendents. It would be desirableto describe the asymptotics via the monodomy data as given in Theorem 8 and Theorem 9.However, to the best of our knowledge the connection between the parameters in the asymptoticbehavior (9.14) for the Painlev´e III transcendents and the monodomy data has not been workedout in the literature. We expect that such a connection formula could be derived, perhaps byusing the Riemann-Hilbert method or Isomonodromy method [12]. This, together with Theorem10, would then give us a description of the asymptotics of the accessory parameters for DHEvia the monodomy data. We will leave this problem to a future consideration.
Acknowledgements
The authors are very grateful to the anonymous reviewers for their constructive comments andsuggestions. The work of Shuai-Xia Xu was supported in part by the National Natural ScienceFoundation of China under grant numbers 11571376 and 11971492. Yu-Qiu Zhao was supportedin part by the National Natural Science Foundation of China under grant numbers 11571375and 11971489. 40 eferences [1] J.B. Amado, B.C. da Cunha and E. Pallante, On the Kerr-AdS/CFT correspondence,
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