On the Logarithmic Asymptotics of the Sixth Painlevé Equation Part I

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1 On the Logarithmic Asymptotics of the Sith Painlevé Equation Part I Davide Guzzetti Abstract We study the solutions of the sith Painlevé equation with a logarithmic asymptotic behavior at a critical point. We compute the monodromy group associated to the solutions by the method of monodromy preserving deformations and we characterize the asymptotic behavior in terms of the monodromy itself. This is the first of two papers aimed at the characterization/classification of the logarithmic behaviors, in terms of the monodromy data. Introduction We consider the sith Painlevé equation: d y d = y + y + dy y d + + dy y d yy y + α + β y + γ + δ y y, PVI. The generic solution has essential singularities and/or branch points in,,. It s behavior at these points is called critical. Other singularities which may appear are poles and depend on the initial conditions. A solution of PVI can be analytically continued to a meromorphic function on the universal covering of P \{,, }. For generic values of the integration constants and of the parameters α,β,γ,δ, it cannot be epressed via elementary or classical transcendental functions. For this reason, it is called a Painlevé transcendent. Solving PVI means: i Determine the critical behavior of the transcendents at the critical points =,,. Such a behavior must depend on two integration constants. ii Solve the connection problem, namely: find the relation between couples of integration constants at =,,. PVI is the isomonodromy deformation equation of a Fuchsian system of differential equations 8: dψ A = A, λ Ψ, A, λ := + A dλ λ λ + A, λ C. λ The matrices A i depend on in such a way that there eists a fundamental matri solution Ψλ, such that its monodromy does not change for small deformations of. They also depend on the parameters α, β, γ, δ of PVI through more elementary parameters θ, θ, θ, θ, according to the following relations: A := A + A + A = θ σ 3, θ. Eigenvalues A i = ± θ i, i =,, ; α = θ, β = θ, γ = θ, δ = θ 3 Here σ 3 := is the Pauli matri. The condition θ is not restrictive, because θ = is equivalent to θ =. The equations of monodromy preserving deformation Schlesinger equations, can be written in Hamiltonian form and reduce to PVI, being the transcendent y the solution λ of A, λ, =. Namely: y = A A + A A, 4

2 The matrices A i, i =,,, depend on y, dy d and y through rational functions, which are given in 8 and in subsection 8.. This paper, and the second paper, are devoted to the computation of the monodromy group of associated to the solutions with a logarithmic critical behavior, and to the action of the symmetries of PVI on the monodromy of. They are part of a project to classify the critical behaviors in terms of the monodromy data of the system. This project has been the motivation of our papers 4. In our paper 4, we developed a constructive procedure which we called matching. It enabled us to compute the leading term of the critical behavior of a transcendent y and the monodromy data of when the matrices A i are those associated to y. Originally, such an approach was suggested by Its and Novokshenov in 5, for the second and third Painlevé equations. The method of Jimbo 7 can be regarded as a matching procedure. This approach was further developed and used by Kapaev, Kitaev, Andreev, and Vartanian see for eample the case of the fifth Painlevé equation, in. Our approach in 4 is new, because we introduced non-fuchsian systems associated to PVI in the process of matching. In this way we obtained new asymptotic behaviors. The matching procedure will be reviewed in section. We developed the matching procedure in order to discover new critical behaviors and to classify the critical behaviors themselves in terms of associated monodromy data. Denote by M, M, M a monodromy representation of. The critical behaviors associated to monodromy matrices satisfying the relation trm i M j ±, i j {,, }, is known from the work 7. But when trm i M j = ±, we cannot naively etend the procedure of 7. In addition, many cases corresponding to non generic values of α, β, γ, δ are not yet studied. The matching procedure was developed in 4, as a general method to study the cases trm i M j = ± and the non generic cases of α, β, γ, δ. The logarithmic solutions, some of the Taylor s series solutions and the trigonometric solutions of 4 actually appear when trm i M j = ± for some i j =,,. The values of the traces trm M, trm M, trm M characterize the critical behaviors at =,, respectively. This is a known fact, which follows from the solution of the connection problem see also subsection 8.3. For eample, in the generic case studied in 7 we find the following behaviors at the critical points 7335: a σ + O ɛ,, y = y = a σ + O ɛ,, y = a σ + O ɛ,, where ɛ is a small positive number, a, σ, a, σ, a, σ are comple numbers such that a, a i and < Rσ <, < Rσ <, < Rσ <. The connection problem among the three sets of parameters a, σ, a, σ, a σ was first solved in 7 and its solution implies that: cosπσ = trm M, cosπσ = trm M, cosπσ = trm M ; while a, a, a are rational functions of the trm i M j s i j =,, and depend on the θ ν s ν =,,, through trigonometric functions and Γ-functions rationally combined. In this sense, the three traces determine the critical behavior at the three critical points. Before we present the result of the paper, it is worth summarizing the results obtained by the matching procedure in 4. We first consider the point =. Let σ be a comple number defined by: trm M = cosπσ, Rσ. The matching procedure yields the following behaviors for : y a σ, if Rσ > ; 5 y {ia sin iσ ln + φ + θ θ + } σ σ, if Rσ =, σ. In the above formulae, σ is one of the integration constants, while a, or φ, is the other. A is: θ θ A := σ θ + σ σ.

3 As we mentioned, the behavior 5 was first studied in 7. For special values of σ, the first leading term above is zero and we need to consider the net leading terms: y θ θ + θ y θ θ θ r θ + θ +σ, σ = ±θ + θ, r θ θ +σ, σ = ±θ θ. When σ =, the matching procedure of 4 yields the logarithmic behaviors: { θ y θ ln + 4r + θ } 4 θ + θ θ θ, if θ θ θ, 6 y r ± θ ln, if θ = θ. 7 Here r is an integration constant. In 4 we also computed all the solutions with Taylor epansions at a critical point. They fall within three equivalent classes the equivalence relations are Backlund transformations of PVI, with representatives characterized by σ = ±θ ± θ,, respectively. To these classes, we must add the singular solutions y =,,. The associated monodromy groups are characterized by reducible subgroups generated by M M and M. Taylor solutions are studied also in 9, by the isomonodromy deformation method; and in by a power geometry technique. The critical behaviors at =, can be obtained from those at = by the action of some of the Backlund transformations of PVI. See subsection 8.3. The monodromy data for the solution 5 are computed in 733. The monodromy data for the Taylor epansions are computed in 4 and 9. In 4 we did not compute the monodromy associated to the logarithmic behaviors, postponing this problem to the present paper and its companion paper in preparation. We are going to show that logarithmic critical behaviors at = are associated to trm M = ±, at = to trm M = ±, and at = to trm M = ±. Once the monodromy data are known, the connection problem is solved see subsection 8.3 We computed the logarithmic asymptotic behaviors in 4 as a result of the matching procedure in the framework of the method of monodromy preserving deformations. In , A.D.Bruno and I.V.Goryuchkina constructed the asymptotic epansions, including logarithmic ones, by a power geometry technique 9. By this technique, the authors of 7 claim that they have obtained all the critical behaviors for PVI. The logarithmic asymptotics for real solutions of PVI is studied in 4. Our approach, being based on the method of isomonodromy deformations, allows to solve the connection problem, while the results of 4 8 and 4 are local.. Results In this paper: In Section 3 we justify the project of classifying the transcendents in terms of monodromy data of. We establish the necessary and sufficient conditions such that there eist a one to one correspondence between a set of monodromy data of system and a transcendent of PVI. The result is Proposition. The definition of monodromy data itself is given in Section 3. We compute the monodromy data associated to the logarithmic solutions 6 in the generic case θ, θ, θ, θ Z. The result is Proposition, Section 5. In particular, trm M =. 3 In Proposition 3 of Section 6, we compute the monodromy group associated to the solution 7. In particular, trm M =. The parameter r will be computed as a function of the θ ν s, ν =,,, and of trm M. 4 We consider a non generic case of 6, which occurs when: θ = θ =, θ =, θ = p, p Z. 8 3

4 Therefore: y p ln + r + p p,. 9 The monodromy of the associated system is computed in Proposition 4, Section 7. It is important to observe that the monodromy is independent of r. This means that the parameter r cannot be determined in terms of the monodromy data. Therefore, 9 is a one parameter class of solutions parameter r associated to the same monodromy data. We prove in Proposition 4 that the solution 9 is associated to: trm M =, trm M =, trm M =. This special values of the traces imply that the behavior at = and = is also logarithmic. trm M = is associated to the logarithmic behavior of type ln at =. trm M = is associated to the logarithmic behavior of type ln / at =. trm M = is associated to the logarithmic behavior of type / ln at =. Actually a solution 9 has the following behaviors at the three critical points: p ln + ρ,, y p ln + ρ,, where: ρ = p ln +ρ,. r + p p, ρ = π4 ln + ρ π i4 ln + ρ ln +, ρ π = ln +. 4 ln + ρ The behavior at = differs from those at =, for the inverse of ln appears. This is actually due to the fact that trm M =. We will prove the above behaviors in section 8.4, and in the second paper by a different method. In general, the logarithmic behaviors of type 6 at the critical points are as follows: { θ y θ ln + 4r + θ } 4 θ + θ θ θ,. θ y y θ θ + θ θ ln θ 4 + 4r + θ θ,. θ { θ θ θ 4 y = θ θ ln + θ θ 4 ln + 4r + θ } θ,. 3 θ + 8r + 4θ 4 θ θ ln + O ln,. 4 4 y = θ θ 8r + 4θ ln θ θ ln + O 4 y = + θ θ ln 8r + 4θ θ θ In general, the log-behaviors of type 7 are: ln + O ln ln,. 5,. 6 y r ± θ ln,, θ = θ. 7 4

5 y r ± θ ln,, θ = θ. 8 y r ± θ ln,, θ = θ. 9 y y = ± θ ln y = ± r ± θ ln,, θ = θ. r θ ln + O ln.,, θ = θ. r θ ln θ ln + O ln,, θ = θ. The above are proved in Section 8, making use of the Backlund transformations of PVI. The behaviors, 7 are associated to trm M = ;, 8 are associated to trm M = ; 3, 9 are associated to trm M =. This fact is proved in Section 8.3. The behaviors 4, are associated to trm M = ; 5, are associated to trm M = ; 6, are associated to trm M =. This fact is proved in the second paper. We note that generically a solution 6 does not have the logarithmic behavior at =,, because the traces trm M, trm M are not equal to ±. The case 9 is special, in that the log-behavior appears at the three critical points. Acknowledgments: The author is supported by the Kyoto Mathematics COE fellowship at RIMS, Kyoto University, Japan. Matching Procedure This section is a review of the matching procedure of 4. We eplain how the asymptotic behavior of a transcendent is derived, and how the associated monodromy is computed.. Leading Terms of y We consider. We divide the λ-plane into two domains. The outside domain is defined for λ sufficiently big: λ δout, δ OUT >. 3 Therefore, can be written as: dψ dλ = A + A + A λ λ n A + λ λ n= The inside domain is defined for λ comparable with, namely: Therefore, λ as, and we rewrite as: dψ dλ = A λ + A λ A Ψ. 4 λ δin, δ IN >. 5 λ n Ψ. 6 If the behavior of A, A and A is sufficiently good, we epect that the higher order terms in the series of 4 and 6 are small corrections which can be neglected when. If this is the case, 4 and 6 reduce respectively to: dψ OUT dλ = A + A + A λ λ N OUT 5 n= λ n= n A + λ Ψ OUT, 7

6 dψ IN dλ = A λ + A N IN λ A λ n n= Ψ IN, 8 where N IN, N OUT are suitable integers. The simplest reduction is to Fuchsian systems: dψ OUT A + A = + A Ψ OUT, 9 dλ λ λ dψ IN dλ = A λ + A Ψ IN. 3 λ In 4 we considered reduced non-fuchsian systems for the first time in the literature, where the fuchsian reduction has been privileged. We showed that in some relevant cases it cannot be used, being the non-fuchsian reduction necessary. Generally speaking, we can parameterize the elements of A + A and A of 9 in terms of θ, the eigenvalues of A + A and the eigenvalues θ of A + A + A. We also need an additional unknown function of. In the same way, we can eplicitly parameterize the elements of A and A in 3 in terms of θ, θ, the eigenvalues of A + A and another additional unknown function of. Cases when the reductions 7 and 8 are non-fuchsian deserve particular care, as it has been done in 4. Our purpose is to find the leading terms of the unknown functions when, in order to determine the critical behavior of A, A, A and of 4. The leading term can be obtained as a result of two facts: i Systems 7 and 8 are isomonodromic. This imposes constraints on the form of the unknown functions. Typically, one of them must be constant. ii Two fundamental matri solutions Ψ OUT λ,, Ψ IN λ, must match in the region of overlap, provided this is not empty: Ψ OUT λ, Ψ IN λ,, δout λ δin, 3 This relation is to be intended in the sense that the leading terms of the local behavior of Ψ OUT and Ψ IN for must be equal. This determines a simple relation between the two functions of appearing in A, A, A, A + A. 3 also implies that δ IN δ OUT. Practically, to fulfill point ii, we match a fundamental solution of 7 for λ, with a fundamental solution of the system obtained from 8 by the change of variables µ := λ/, namely with a solution of: dψ IN dµ = A µ + A µ A N IN n= n µ n Ψ IN, µ := λ. 3 To summarize, matching two fundamental solutions of the reduced isomonodromic systems 7 and 8, we obtain the leading terms, for, of the entries of the matrices of the original system. The procedure is algorithmic, the only assumption being 3. This method is sometimes called coalescence of singularities, because the singularity λ = and λ = coalesce to produce system 7, while the singularity µ = and µ = coalesce to produce system 3. Coalescence of singularities was first used by M. Jimbo in 7 to compute the monodromy matrices of for the class of solutions of PVI with leading term y a σ, < Rσ <.. Computation of the Monodromy Data In the λ-plane C\{,, } we fi a base point λ and three loops, which are numbered in order,, 3 according to a counter-clockwise order referred to λ. We choose,, to be the order,, 3. We denote the loops by γ, γ, γ. See figure. The monodromy matrices of a fundamental solution Ψλ w.r.t. this base of loops are denoted M, M, M. The loop at infinity will be γ = γ γ γ, so M = M M M. As a consequence, the following relation holds: cosπθ trm M + cosπθ trm M + cosπθ trm M 6

7 λ 3 order,, 3. γ γ γ Figure : The ordered basis of loops = cosπθ + 4 cosπθ cosπθ cosπθ. The monodromy matrices are determined by trm ν, trm ν M µ, ν, µ =,,, 3. As a consequence of isomonodromicity, there eists a fundamental solution Ψ OUT of 7 such that M OUT = M, M OUT = M, where M OUT and M OUT are the monodromy matrices of Ψ OUT at λ =,. Moreover, M OUT = M M. There also eists a fundamental solution Ψ IN of 8 such that: M IN = M, M IN = M, where M IN and M IN are the monodromy matrices of Ψ IN at λ =,. The method of coalescence of singularities is useful when the monodromy of the reduced systems 7, 8 can be eplicitly computed. This is the case when the reduction is fuchsian namely 9, 3, because fuchsian systems with three singular points are equivalent to a Gauss hypergeometric equation see Appendi. For the non-fuchsian reduction, in general we can compute the monodromy when 7, 8 are solvable in terms of special or elementary functions. In order for this procedure to work, not only Ψ OUT and Ψ IN must match with each other, as in subsection., but also Ψ OUT must match with a fundamental matri solution Ψ of in a domain of the λ plane, and Ψ IN must match with the same Ψ in another domain of the λ plane. The standard choice of Ψ is as follows: I + O λ λ θ σ3 λ R, λ ; ψ I + Oλ λ θ σ 3 λ R C, λ ; Ψλ = ψ I + Oλ λ θ σ3 λ R C, λ ; ψ I + Oλ λ θ σ 3 λ R C, λ ; 33 7

8 Here ψ, ψ, ψ are the diagonalizing matrices of A, A, A respectively. They are defined by multiplication to the right by arbitrary diagonal matrices, possibly depending on. C ν, ν =,,,, are invertible connection matrices, independent of 8. Each R ν, ν =,,,, is also independent of, and:, if θ ν > integer R ν = if θ ν Z, R ν =, if θ ν < integer If θ i =, i =,,, then R i is to be considered the Jordan form of A i. Note that for the loop λ λe πi, λ > ma{, }, we immediately compute the monodromy at infinity: M = ep{ iπθ } ep{πir }. Let Ψ OUT and Ψ IN be the solutions of 7 and 8 matching as in 3. We eplain how they are matched with 33. * Matching Ψ Ψ OUT : λ = is a fuchsian singularity of 7, with residue A /λ. Therefore, we can always find a fundamental matri solution with behavior: Ψ Match OUT = I + O λ λ θ σ3 λ R, λ. This solution matches with Ψ. Also λ = is a fuchsian singularity of 7. Therefore, we have: Ψ Match OUT = ψ OUT I + Oλ λ θ σ 3 λ R C OUT, λ ; Here C OUT is a suitable connection matri. ψ OUT is the matri that diagonalizes the leading terms of A. Therefore, ψ ψ OUT for. As a consequence of isomonodromicity, R is the same of Ψ. As a consequence of the matching Ψ Ψ Match, the monodromy of Ψ at λ = is: OUT M = C ep{iπθ σ 3 } ep{πir }C, with C C OUT. We finally need an invertible connection matri C OUT to connect Ψ Match OUT appearing in 3. Namely, Ψ Match OUT = Ψ OUT C OUT. * Matching Ψ Ψ IN : with the solution Ψ OUT As a consequence of the matching Ψ Ψ Match OUT, we have to choose the IN-solution which matches with Ψ Match. This is ΨMatch IN := Ψ IN C OUT. OUT Now, λ =, are fuchsian singularities of 8. Therefore: Ψ Match IN = ψ IN I + Oλ λ θ σ 3 λ R C IN, λ ; ψ IN I + Oλ λ θ σ3 λ R C IN, λ ; The above hold for fied small. Here C IN and C IN are suitable connection matrices. ψ IN and ψ IN are diagonalizing matrices of the leading terms of A and A. For they match with ψ and ψ of Ψ in 36. On the other hand, as a consequence of isomonodromicity, the matrices R and R are the same of Ψ. The above Ψ Match IN has the same behavior of Ψ at λ and λ ; moreover, it is an approimation of Ψ for small. The matrices C IN, C IN are independent of. So, the matching Ψ Ψ IN is realized and the connection matrices C and C coincide with C IN, C IN respectively. As a result, we obtain the monodromy matrices for Ψ: M = C ep{iπθ σ 3 } ep{πir }C, 8 C C IN,

9 M = C ep{iπθ σ 3 } ep{πir }C, C C IN. Our reduction is useful if the connection matrices C OUT, C IN, C IN can be computed eplicitly. This is possible for the fuchsian reduced systems 9, 3. For non-fuchsian reduced systems, we discussed the computability in 4. 3 Classification in Terms of Monodromy Data Two conjugated systems: dψ d = A, λ Ψ, Ψ = Ã, λ Ψ, dλ dλ Ψ = W Ψ, detw, Ã = W AW, admit fundamental matri solutions with the same monodromy matrices w.r.t. the same basis of loops. The matri Ã, λ defines the same solution of PVI associated to A, λ only if the following condition holds: Ã + Ã + Ã = θ σ 3, where Ãi = W A i W, i =,,. Namely, W σ 3 W = σ 3. This occurs if and only if W is diagonal. The transformation of A, λ is therefore: W A, λw w A, λ = w A, λ w w, where W =. w A, λ A, λ w We conclude that the equation A, λ = is the same and then: Two conjugate fuchsian systems, satisfying 3, define the same solution of PVI if and only if the conjugation is diagonal. Note that θ is a necessary condition, otherwise any W would be acceptable and then A, λ = would not define y uniquely. The problem of finding a branch of a transcendent associated to a monodromy representation is the problem of finding a fuchsian system having the given monodromy. This problem is called Riemann-Hilbert problem, or th Hilbert problem. For a given PVI there is a one-to-one correspondence between a monodromy representation and a branch of a transcendent if and only if the Riemann-Hilbert problem has a unique solution A, λ, defined up to diagonal conjugation. Riemann-Hilbert problem R.H.: find the coefficients A i, i =,, from the following monodromy data: a A fied order of the poles,,. Namely, we choose a base of loops. Here we choose the order,,3=,,. See figure. b The eponents θ, θ, θ, θ, with θ. c Matrices R, R, R, R, such that:, if θ ν > integer R ν = if θ ν Z, R ν =, if θ ν < integer R j =, if θ j =, j =,,. c three monodromy matrices M, M, M relative to the loops, similar to the matrices ep{iπθ i σ 3 } ep{πir i }, i =,,, satisfying for the chosen order of loops γ γ γ = γ : M M M = e iπθ σ3 e πir 9

10 Solving the Riemann-Hilbert problem means that we have to find invertible connection matrices, C ν, ν =,,,, such that. C j e iπθjσ3 e πirj C j = M j, j =,, ; 34 C e iπθ σ3 e πir C = e iπθ σ3 e πir. 35 and a matri valued meromorphic function Ψ, λ such that: I + O λ λ θ σ3 λ R C, λ ; ψ I + Oλ λ θ σ 3 λ R C, λ ; Ψ, λ = ψ I + Oλ λ θ σ3 λ R C, λ ; ψ I + Oλ λ θ σ 3 λ R C, λ ; 36 Here ψ, ψ, ψ are invertible matrices depending on. The coefficient of the fuchsian system are then given by dψ, λ A; λ := Ψ; λ. dλ A R.H. is always solvable at a fied. As a function of, the solution A; λ etends to a meromorphic function on the universal covering of C\{,, }. Now we prove the following fact: The R.H. admits diagonally conjugated solutions fuchsian systems, ecept when at least one θ ν Z\{} and simultaneously R ν =. This can be equivalently stated in the form of the following: Proposition There is a one to one correspondence between the monodromy data θ, θ, θ, R, R, R, θ, R, M, M, M defined up to conjugation, satisfying a, b, c above, and a branch of a transcendent y, ecept when at least one θ ν Z\{} and simultaneously R ν =. To say in other words, the one to one correspondence is realized if and only if one of the following conditions is satisfied: θ ν Z, for every ν =,,, ; if some θ ν Z and R ν, θ ν 3 if some θ j = j =,, and simultaneously θ Z, or θ Z and R. πi Note that for θ j =, M j can be put in Jordan form. Therefore Proposition says that: There is one to one correspondence ecept when one of the matrices M i i =,,, or M = M M M, is equal to ±I. Proof: The proof is based on the observation that a triple of monodromy matrices M, M, M may be realized by two fuchsian systems which are not conjugated. The crucial point is that the solutions of 34, 35 are not unique. Two sets of particular solutions C ν and C ν ν =,,, give to fuchsian systems: dψ, λ Ψ, λ = A, λ, dλ d Ψ, λ dλ Ψ, λ = Ã, λ. These may be not diagonally conjugated. If this happens, there is no one-to-one correspondence between a set of monodromy data and a solutions of PVI. We study the structure of the solutions of 34, 35. Equation 35 has the following solutions:

11 i If θ Z and then R =, p C = q ii If θ Z and R, p q C = p where p, q C, p. C =, p, q C\{}, if R = p, if R q p = iii If θ Z and R =, then C is any invertible matri.. Equation 34, may have different solutions C j and C j. Therefore C j C j Cj C j e iπθ jσ 3 e πirj C C j j = e iπθjσ3 e πirj. i If θ j Z and then R j =, we have: C C aj j j =, b j a j, b j C\{} ii If θ j Z and R j, we have: C C aj b j j = j, a a j, b j C, a j ; if R j = j C C aj j j =, a b j a j, b j C, a j ; if R j = j In particular, for θ j =, R j is the Jordan form iii If θ j Z and R j =, then C j C j. a b is any invertible matri c d is a solution of:. Let C ν and C ν ν =,,, be two sets of solutions of and let us denote by Ψ and Ψ the corresponding solutions of the R.H. We observe that: i for θ j Z j =,, : λ j θ j aj aj σ3 = λ j θ j σ3. b j b j or ii For θ j Z and R j : λ j θ j aj b σ3 λ j Rj j = a a j I + λ j θj bj j λ j θ j aj σ3 λ j Rj = for R j upper or lower triangular respectively. b j iii For θ j Z and R j = : λ j θ j a b a bλ θ j σ3 = c d cλ θj d a j. λ j θ j σ3 λ j Rj, a j I + λ j θj λ j θ j σ3 λ j Rj, b j λ j θ j σ3

12 We conclude that, for λ j: aj, if θ b j Z; j Ψ Ψ a j I, if θ j, aj b j, if θ a j =, j Arbitrary invert. matri, if θ j =, C λ j θj, otherwise, The matri C above is C = or C =. if θ j Z, R j if θ j Z, R j = Let C and C be two solutions of 35. i If θ Z and then R =, we have C C a = b ii If θ Z and R, we have C C = a b a C C b a = a iii If θ Z and R =, then C C, a, b C\{}., a, b C, a ; if R =, a, b C, a ; if R = is any invertible matri.. Therefore, for λ we have: a if θ b Z; Ψ Ψ I + O λ a I + b λ a I, if θ θ Z\{}, R C λ θ, if θ Z\{}, R = The matri C above is C = or C =. From the above result we conclude that Ψ Ψ is analytic on C and then it is a constant matri W, ecept when at least one θ ν Z\{} and simultaneously R ν =. Ecept for this case, we have: Ψ = W Ψ = Ã, λ = W A, λw. We observe that: W = lim λ Ψ Ψ in the cases θ Z, or for θ Z θ and R. Therefore W is diagonal. Proposition is proved. 4 Logarithmic asymptotics 6 and 7 We consider cases when can be reduced to the fuchsian systems 9 and 3. comple number defined, up to sign, by: Let σ be a tr M M = cosπσ, Rσ.

13 In our paper 4, we computed all the asymptotic behaviors for Rσ <, as they can be obtained from the matching procedure when 9 and 3 are fuchsian. Among them, we obtained 6 and 7. Note: For solutions with epansion: y = A + B ln + C ln + D ln A + B ln ,. only the following cases are possible: θ θ ±θ + O Taylor epansion, y = θ B + B θ ln + θ θ ln , θ 4 A ± θ ln , and θ = ±θ. 37 A and B are parameters. We see that the higher orders in 6 and 7 are O ln m, for some integer m >. 4. Review of the Derivation of 6 and 7 Let. The reduction to the fuchsian systems 9 is possible if in the domain 3 we have: A + A ij A ij, namely: A + A ij A ij δout. 38 λ Let us denote with Âi the leading term of the matri A i, i =,,. We can substitute 9 with: dψ OUT  + =  + Â Ψ OUT 39 dλ λ λ Lemma If the approimation 9 is possible, then  + has eigenvalues ± σ C independent of, defined up to sign and addition of an integer by trm M = cosπσ. Let r C, r. For θ, the leading terms are: σ θ θ 4θ r  = σ θ θ σ θ +θ 6θ r σ θ θ 4θ, 4 and  + ˆ A = θ σ θ 4θ r σ θ θ σ θ +θ 6θ r θ σ θ 4θ. 4 Proof: Observe that trâ +  = tra + A =, thus, for any,  +  has eigenvalues of opposite sign, that we denote ± σ/. Then, we recall that is a monodromy preserving deformation, therefore the monodromy matrices of 39 are independent of. At λ =,, they are: { M OUT M M =, M OUT = M M M, M OUT = M. Thus, detm OUT =, because detm =detm =. Therefore, there eists a constant matri D and a comple constant number σ such that: diagep{ iπσ}, ep{iπσ}, D M OUT D = ± ±, or, σ Z ± ± We conclude that σ σ. We also have trm OUT = cosπσ. 3

14 Now consider the gauge: Φ := λ σ λ θ Ψ OUT. dφ dλ =  +  σ λ +  θ λ Φ 4 We can identify  +  σ and  θ with B and B of Proposition 5 in Appendi, case 69, with a = θ + θ + σ, b = θ + θ + σ, c = σ. In principle, r may be a function of. If the monodromy of system 39 depends on r, then r is a constant independent of. This is the case here. For all the computations which follow, involving system 39 or 4, we note that the hypothesis θ ecludes cases 7, 7 and the Jordan cases The reduction to the fuchsian system 3 is possible for in the domain 5 if: A ij + A ij λ λ A ij, namely: A + A ij A ij. 43 δin We can rewrite 3 using just the leading terms of the matrices: dψ IN dλ =  λ + A ˆ Ψ IN, 44 λ Then, we re-scale λ and consider the following system: dψ IN  dµ = µ + Â Ψ IN, µ µ := λ We know that there eists a matri K such that: σ K  +  K = σ, or. Let  i := K  i K, i =,. By a gauge transformation, we get the system: Ψ IN =: K Ψ, dψ dµ =  µ + A ˆ Ψ, 45 µ Important Remark see 4: Conditions 38, 43 are satisfied if and only if Rσ <, < δ IN δ OUT <. 4. Matching for σ =. Proof of 6 and 7 We suppose now σ =. 4.. Case θ ± θ. Proof of 6 Lemma Let r C, r. The matrices of system 39 are: θ +θ 4θ  = r θ θ 6θ r θ +θ 4θ θ,  + A ˆ θ 4θ = r θ θ 6θ r θ θ 4θ, r. A fundamental matri solution can be chosen with the following behavior at λ = : log λ Ψ OUT λ = G + Oλ, G = θ θ. 4θ r r 4

15 Proof: The system 4 is: dφ dλ =  +  λ +  θ λ Φ, We identify  +  and  θ with B and B of proposition 5 in Appendi, diagonalizable case 69 we recall that 7 74 never occur when θ with a = θ + θ, b = θ + θ, c =. The behavior of a fundamental solution is a standard result in the theory of Fuchsian systems. The matri G is defined by G  + ˆ A G =. Lemma 3 Let r C. The matrices of system 45 are: r + θ 4 r r+θ θ r θ  = θ, ˆ 4 r r+θ θ A θ = θ θ 4 r θ θ θ 4 r + θ.. 46 There eist a fundamental solution of 45 with the following behavior at µ = : log µ Ψ µ = I + O, µ. µ Proof: We do a gauge transformation: Φ := µ θ µ θ Ψ, We identify  θ, A ˆ θ parameters a = θ + θ, c = θ. In particular, dφ dµ = Â θ µ + ˆ A θ µ Φ. 47 with B and B in the Appendi, Proposition 5, case 7, with  θ + A ˆ θ = θ +θ θ+θ Here the values of the parameters satisfy the conditions a and a c, namely θ ± θ. From the matrices 7, we obtain  = B + θ / and A ˆ = B + θ /. Keeping into account 48, by the standard theory of fuchsian systems we have: Φ µ = I + O µ θ +θ log µ, µ. µ This proves the behavior of Ψ µ. If the monodromy of the system 45 depends on r, then r is a constant independent of. This is the case here. The matching condition Ψ OUT λ K Ψ λ/ becomes: log λ K log λ G = K θ θ 4 θ r r log. From the above result, together with 46, we compute  = K  K,  = K  K. For eample,  = G r + θ + θ θ θ log θ log r + θ 4 4 log + 4 rr+θ θ θ θ θ θ θ log G. r + θ 4 4 A similar epression holds for entries  = r and:  = r θ θ 4 48 ˆ A. The leading terms of y are obtained from 4 with matri log r + θ log + 4 rr + θ θ θ. 5

16 The result is: = The above is 6. y θ θ 4 { θ θ log 4 log r + θ r + θ log + log + 4 rr + θ θ θ 4 θ θ r + θ θ 4 } Case θ ± θ =. Proof of 7 We consider here the cases 73, 74 of Proposition 5 applied to the system 47. Case 73 is the case σ =, θ = θ, with a =, c = θ in the system 47. From Proposition 5 we immediately have: θ Â = r θ, ˆ r A θ =. θ The behavior of Ψ and Ψ OUT, and the matching are the same of subsection 4... We obtain the same K. Therefore: Â = r r θ ln, Â = r. This gives the leading terms: y r θ ln = r + θ ln. 5 In the same way, we treat the other cases. Case 73 with a = c, is the case σ =, θ = θ. As above, we find y r θ ln = r θ ln. Case 74 with a =, is the case σ =, θ = θ. We find y r + θ ln = r θ ln. Case 74 with a = c, is the case σ =, θ = θ. We find y r + θ ln = r + θ ln. Both 49 and 5 contain more than one term, and in principle only the leading one is certainly correct. To prove that they are all correct, we observe that 49 and 5 can be obtained also by direct substitution of y = A + B ln + C ln + D ln A + B ln into PVI. We can recursively determine the coefficients by identifying the same powers of and ln. As a result we obtain only the five cases 37, which include 49 and 5. The reader can verify that conditions 38, 43 are satisfied. 5 Monodromy Data associated to the solution 6 In this section, we compute the monodromy data for the solution 6 in the generic case θ ν Z for any ν =,,,. We need some notations. Let γ E denote the Euler s constant. Let: In particular, ψ E = γ E. ψ E = d ln Γ,,,, 3,... d Proposition Let θ, θ, θ, θ Z. The monodromy group associated to 6 is generated by: M = EC ep{iπθ σ 3 } M = EC C ep{iπθ σ 3 } C EC, EC, The matrices above are: M = BC ep{iπθ σ 3 } C B. E = 4q θ θ 4 θ θ 4 θ θ 6

17 q = 4iπɛ + + θ θ where ɛ = ±. C = { 4r + θ θ + θ θ B = sin π θ θ C = C = θ eiπɛ + θ Γ θ + θ sin π θ + θ e iπɛ θ + θ With the above choice, we have: where: ψ θ θ θ + ψ θ } + + γ E, Γ+θ Γ θ θ π sin πθ Γ+θ θ eiπɛ Γ θ θ Γ θ + θ Γ θ θ Γ θ Γ+θ Γ θ θ θ Γ θ θ Γ θ θ θ Γ θ θ Γθ Γ+θ Γθ θ + θ Γ θ + θ Γ θ θ + θ Γ θ + θ Γ θ Γ θ θ Γ Γ + θ θ Γθ Γ θ θ Γ + θ θ θ + θ θ Γ, Γ θ + θ, Γ+θ Γ θ θ θ ω θ, ω := ψ E + θ θ ψ E θ + + γ E, M M M = C OUT C OUT = BC D, D = We also note that trm M =. θ r ep{ iπθ σ 3 } C OUT,, C = π e i π θ +θ sin π θ+θ π e i π θ θ sin π θ θ If we compute trm M and trm M we find two quadratic polynomials of q. Then, q can be derived as a function trm M and trm M. In this way we obtain r = rθ, θ, θ, θ, trm M, trm M 5 We omit the long formula which results. Direct computation shows also that trm M and trm M depend on ɛ only through q. Therefore, different choices of ɛ just change the branch of 6, because they change 4r/θ θ of 8πi. 5. Derivation of Proposition The matching Ψ OUT Ψ IN has been realized by: log λ Ψ OUT, λ = G + Oλ, Ψ µ = Ψ IN, λ = K Ψ λ, G = I + O µ θ θ. 4θ r r log µ, µ. MATCHING Ψ Ψ OUT. The correct choice of Ψ Match OUT must match with: Ψ = I + O λ λ θ σ3, λ. 7

18 System 4 is 69 of Appendi, with: If we write: a = θ + θ, b = θ + θ, c =. Ψ OUT = λ θ ϕ ϕ, ξ ξ then ϕ and ϕ are independent solutions of the hypergeometric equation 75: while ξ i are given by 76: λ λ d ϕ dλ + + c a + b + + λ dϕ ab + ϕ =, dλ ξ i = r λ λ dϕ i dλ a λ + b c a b ϕ i, i =,. We need a complete set of solutions at λ =,,. We eplain some preliminary facts. Let us consider a Gauss hypergeometric equation in standard form: z z d ϕ dz + γ α + β + z dϕ dz α β ϕ = 5 α, β, γ here are not the coefficients of PVI! We are just using the same symbols only here. We refer to the paper by N.E. Norlund in order to choose three sets of two independent solutions which can be easily epanded in series at z =,, respectively. Solutions with logarithmic or polynomial behaviors at z = may occur when γ Z. The role of γ at z = and z = is played by α + β γ + and α β + respectively. Therefore, solutions with logarithmic or polynomial behaviors at z = may occur when α + β γ + Z, at z = when α β + Z. Some more words must be said about the choice of independent solutions. We consider the point z =. For γ Z, we choose the following two independent solutions: ϕ z = F α, β, γ ; z, ϕ z = z γ F α, β, γ ; z. Here F is the standard hypergeometric function and α = α γ +, β = β γ +, γ = γ. If γ =,,,..., then: ϕ z = fα, β, γ ; z, ϕ z = z γ F α, β, γ ; z, if α or β =,,..., γ. ϕ z = z γ Gα, β, γ ; z, ϕ z = z γ F α, β, γ ; z, if α and β,,..., γ. Here f is the truncation of F at the order z γ. G is one of the functions g, g, g or G with logarithmic behavior, introduced in, section. They are listed in Appendi 3. If γ =, 3,..., then: ϕ z = F α, β, γ ; z, ϕ z = z γ fα, β, γ ; z, if α or β =,,..., γ. ϕ z = F α, β, γ ; z, ϕ z = Gα, β, γ ; z, if α and β,,..., γ. If γ =, then: ϕ z = F α, β, γ ; z, ϕ z = Gα, β, γ ; z. The point z = is treated in the same way, with the substitution: α α, β β, γ α + β γ + ; ϕ ϕ, z z. The point z = is treated in the same way, with the substitution: In our case: α α, β α γ +, γ α β + ; ϕ z α ϕ, z z. α = a = θ + θ, β = b + = θ θ +, γ = c + =, z = λ. 8

19 Because γ =, we have a logarithmic solution at λ =. As for λ =, α + β γ + = + θ and for λ =, α β + = θ. We suppose θ and θ Z. We choose the following set of independent solutions at λ =,, respectively the upper label indicates the singularity: { ϕ = F α, β, γ ; λ, ϕ = gα, β, γ ; λ; Let: { ϕ = F α, β, α + β γ + ; λ, ϕ = λ γ α β F γ α, γ β, γ α β + ; λ; { ϕ = λ α F α, α γ +, + α β ; λ, ϕ = λ β F β, β γ +, α + β ; λ ; Ψ i OUT = λ θ From Norlund, 3. and 3. we get: i ϕ ϕ i ξ i ξ i, i =,,. Ψ OUT = Ψ OUT C, arg λ < π, arg λ < π, where C is written in Proposition. From Norlund,. and.3 we obtain: where C is written in Proposition. Ψ OUT = Ψ OUT C, < arg z < π, Note about the computation: In order to apply the formulae of Norlund,. and.3 we have to transform g into g, using the formula see Norlund, formula 4: gα, β, γ; z = g α, β, γ; z π sin πα eiπɛα F α, β, γ; z, 53 where ɛ is an integer introduced as follows. gα, β, γ; z is defined for argz < π, while g α, β, γ; z is defined for arg z < π. Moreover, z = e iɛπ z. In gα, β, γ; z, lnz is negative for < z < namely, argz =, while in g α, β, γ; z, ln z is negative for < z <. Namely, for < z <, we have argz = πɛ. Formula 53 holds true for < arg z < π when ɛ =, and for π < arg z < when ɛ =. In the formulae of Norlund,. and.3 it is required that agr z < π, namely arge iɛπ z < π. This limitation must be restricted to < arg z < π when ɛ =, and for π < arg z < when ɛ = in order to apply 53. In our computations we have chosen < arg z < π i.e. ɛ =, because this is the choice which gives the order M M M = ep{ iπθ σ 3 }. The choice π < arg z < ɛ = gives M M M = ep{ iπθ σ 3 }. We epand ϕ, ϕ in series at λ = and we get: Ψ OUT = G ln λ I + Oλ Be i π θ, λ, where B is written in Proposition. Namely: We epand ϕ, ϕ Ψ OUT = Ψ OUT Be i π θ in series at λ =, obtaining: Ψ OUT I = + O λ where D is written in Proposition. Namely, λ θ σ3 D, λ, Ψ OUT = ΨMatch OUT D. 9

20 Combining the above results we get: Ψ Match OUT = Ψ OUT D = Ψ OUT C D = Ψ OUT BC D e i π θ Ψ OUT C OUT. The matri BC D e i π θ is C OUT. It differs from the matri C OUT of proposition by the factor e i π θ, which simplifies in the formulae. We also have: Ψ Match OUT = Ψ OUT C C D. Finally, it is an elementary computation to see that Ψ OUT = λ θ ϕ ϕ ξ ξ Ψ OUT eiπθσ3, when λ λ e πi. Thus, a choice for the matri M of is M M OUT = DC C eiπθσ3 C C D, = C OUT BC eiπθσ3 C B C OUT. MATCHING Ψ Ψ IN The system: Φ := µ θ µ θ Ψ, dφ dµ = Â θ µ + ˆ A θ µ Φ. is 7 of Appendi, with: a = θ + θ, c = θ. The equation for ξ is in Gauss hypergeometic form 77: µµ d ξ dµ + + c a + µ dξ aa + ξ =, 54 dµ while ϕ is given by 78: ϕµ = µµ dξ + aµ c rξ. aa c dµ In the standard form we have: µ µ d ξ dµ + γ α + β + µ dξ dµ α β ξ =, 55 α = a = θ + θ, β = a + = θ + θ +, γ = c + = θ + ; z = µ. Therefore γ = + θ, α + β γ + = + θ, α β + =, and 54 has no logarithmic solutions at µ =, if θ, θ Z. On the other hand, at µ = we may have a solution with logarithmic or polynomial behavior. For θ, θ Z, we choose the following independent solutions at µ =,, respectively:: { ξ = F α, β, γ ; µ ξ = µ γ F α γ +, β γ +, γ ; µ;

21 { ξ = F α, β, α + β γ + ; µ ξ = µ γ α β F γ β, γ α, + γ α β ; µ; { ξ = µ β g β, γ + β, α + β ; µ ξ = µ β F β, γ + β, α + β ; µ ; Let us construct three fundamental matrices form the above three sets of independent solutions: Ψ i := µ θ µ θ ϕ i ϕ i ξ i ξ i, i =,, The connection formulae between solutions at µ = and is a standard one, and can be found in any book on special functions: Ψ = Ψ C, argµ < π, arg µ < π where C is given in the statement of Proposition. The connection formulae between solutions at µ = and µ = can be found in Norlund, formulae 9. and 9.5 case m =. We get: Ψ = Ψ C, arg µ < π, where C can be read in Proposition and µ = e iπɛ µ when µ <, argµ = πɛ. Note about the computation: In order to apply the formulae 9. and 9.5 of Norlund, we have made use of the formula: g α, β, γ; z = g β, α, γ; z + We epand ξ, ξ, ϕ, ϕ Ψ = where E can be read in Proposition. Thus, π sin πβ α sin πβ sin πα F α, β, γ; z. for µ. We obtain: ln µ I + E, µ µ Ψ = Ψ E, where Ψ is the matri used in the matching Ψ OUT Ψ IN. Epanding ξ, ξ, ϕ, ϕ for µ we get: Ψ = µ θ 4θ+r θ θ 4r θ θ Epanding ξ, ξ, ϕ, ϕ for µ we get: Ψ = The above imply that: Finally, we observe that: θ θ +r θ θ θ +θ +r θ θ + Oµ µ θ σ 3, µ. + O µ µ θ σ3, µ. Ψ Ψ eiπθσ3, for µ µe πi, Ψ Ψ eiπθσ3, for µ µ e πi. Ψ Match IN = Ψ IN C OUT, Ψ IN = K Ψ = K Ψ E = K Ψ C E K Ψ C C E

22 As a result of the matching procedure we get: M M IN = C OUT EC eiπθσ3 C E C OUT, M M IN = C OUT EC C e iπθ σ 3 C C E C OUT. When we come to the computation of the traces, we find: trm M = aq + b aωq + c bω + aω, trm M = Aq + B Aωq + C Bω + Aω, where a, b, c, A, B, C are complicated long trigonometric epressions in sines and cosines of the parameters πθ ν, ν =,,,. We omit to write them. The above form for the system which determines q and therefore r implies that: Moreover: q = ω + { solution of the system for ω = }. { solution of the system for ω = } = a C trm M A c trm M We omit all the eplicit epressions. Ab ab b C trm M B c trm M a C trm M A c trm M 6 Monodromy Data associated to the Solution 7 Proposition 3. The monodromy group associated to the solution 7: y r + θ ln, is generated by: M = E ep{ iπθ σ 3 } E, M = EU ep{iπθ σ 3 } UE, M = BC ep{iπθ σ 3 } C B ; where B, C are given in Proposition and: E := e i π θ r θ Ψ E θ + γ E iπ, U := Γθ + Γ θ. e i π θ Conversely, the parameter r is: + π r θ = π 4 trm M sin πθ sin π θ + θ sin π θ θ + Ψ Eθ + + iπ + γ E + cos πθ + θ sin πθ sin π θ + θ sin π θ θ ω ω is given in Proposition.. The monodromy group and r for the solution 7: cos πθ + θ cos πθ θ sin πθ sin πθ. 56 y r θ ln, are obtained from the results in, with the substitution θ θ.

23 Proof: For the matching Ψ OUT Ψ IN and Ψ Ψ OUT, we proceed as in the proof of Proposition. MATCHING Ψ Ψ IN Consider the case θ = θ. For this case, the system for Φ can be chosen to be 73 or 74, with a = c = θ. Here we refer to system 74. Therefore, a fundamental solution is see Proposition 6: = e i π θ µ θ µ θ Ψ := µ θ θ µ Φ = π θ µ θ µ θ θ µ θ + + µ θ F + θ, + θ, + θ ; µ. µ θ µ θ Here, the branch is: µ = e iπ µ. When µ, we write the hypergeometric function as follows, using the connection formula 9. in Norlund : F + θ, + θ, + θ ; µ = e iπθ θ + µ θ g, + θ, ;, < arg µ < π. µ Here, we have used the branch µ = e iπ µ. The function g is: g, + θ, ; µ From the above, we obtain: Ψ = = Ψ E + θ + γ E + iπ ln µ + + µ ν= ln µ Ee i π θ Ψ Ee i π θ. + θ ν µ ν, µ. ν ν! Here, Ψ is the matri used in the matching Ψ OUT Ψ IN and E is in the statement of the proposition. When µ, we use the connection formula: F + θ, + θ, + θ ; µ = = Γ θ Γ + θ F + θ, + θ, + θ ; µ + Γθ Γ + θ Γ + θ µ θ F,, θ ; µ. Therefore, = e i π θ I + O µ Ψ r θ ΓθΓθ+ θ +Γθ + Finally, when µ,we have: Ψ = e i π r/θ θ + Oµ µ θ σ 3 U, µ. µ θ σ 3. Let C OUT be the same matri introduced in the proof of Proposition. We have: This implies that: Ψ Match IN = Ψ IN C OUT = K Ψ C OUT = K Ψ E C OUT. M = C OUT EU ep{iπθ σ 3 } UE C OUT, M = C OUT E ep{ iπθ σ 3 } E C OUT. The matri C OUT has been simplified in the statement of the proposition. The proof for θ = θ is analogous for eample, it is the case 73 with a =, c = θ. 3

24 7 Monodromy Data for the Non-generic Case 9 We consider the non-generic case θ = p, p Z, θ, θ = θ =, θ =. In this case, the solutions 6 becomes 9. We show here that the solutions 9 are not in one to one correspondence with a set of monodromy data. Namely, to a given set of monodromy data, as defined in Proposition, there corresponds a one parameter family 9, where r is a free parameter i.e. r is not a function of the traces of the product of the monodromy matrices. We miss the one-to-one correspondence because the conditions in Proposition are not realized. Namely, the matri R associated to 9 is: R =, while θ Z and θ. This fact is contained in the following Proposition. Proposition 4 The monodromy group associated to 9 is generated by: πi M = I, M =, M = 8i π ln In particular, trm M = trm M =, The monodromy is independent of the parameter r in 9. i π 3i π trm M = ln. + 8 i π ln Note: With the above choice the monodrmy at infinity: M M or M M is not in standard Jordan form. Namely: M + = M M = 8i π ln i 8i π 4 ln + iπ 3, M π i π 3 + 8i π ln = M M = ln i π 4i ln + π i π + 8i π ln They can be put in Jordan form respectively by the following matrices: C + OUT = 4i π ln 6i π ln r, C i π + 4i π ln OUT = + 4i π ln r i π We obtain: C + OUT On the other hand: M + C + πi OUT = r, C OUT C + OUT M C + OUT = C OUT M C OUT = C + OUT M C + OUT = C OUT M C OUT = 8i π ln i π r 3 8i π ln i π r 6i π ln r 4i π ln r M C πi OUT = r 8i π ln 3i π ln r, i π r + 8i π ln i π 4i ln + π r, 3 + 8i π ln i π 4i ln π r + 8i π ln, r C.. 4

25 7. Derivation of Proposition 4 The matching Ψ OUT Ψ IN has been realized by Ψ OUT, λ = G + Oλ Ψ IN, λ = K Ψ λ log λ, Ψ µ = I + O, G = µ 4r r log µ., µ. MATCHING Ψ Ψ OUT. The correct choice of Ψ Match OUT Ψ = I + O must match with: λ System 4 is 69 of Appendi, with: If we write: λ σ3 λ R, R = a =, b =, c =. ϕ ϕ Ψ OUT = ξ ξ r, λ. then ϕ and ϕ are independent solutions of the hypergeometric equation 75: and λ λ d ϕ dλ + + c a + b + + λ dϕ ab + ϕ =, dλ ξ i = r λ λ dϕ i dλ, a λ + b c a b ϕ i, i =,. We need a complete set of solutions at λ =,,. In the standard Gauss hypergeometric form 5 we have α = β = /, γ =. Since γ =, α + β γ + = and α β + =, we epect solutions with logarithmic behaviors at λ =,,. We choose three sets of independent solutions: { ϕ = F α, β, γ ; λ F,, ; λ, ϕ = gα, β, γ ; λ g,, ; λ ; Let { ϕ = F α, β, α + β γ + ; λ F,, ; λ, ϕ = gα, β, α + β γ + ; λ g,, ; λ ; { ϕ = λ β F β, β γ +, β α + ; λ λ F,, ; λ, ϕ = λ β gβ, β γ +, β α + ; λ λ g,, ; λ ; Ψ i OUT = ϕ i From Norlund, formulae 5. and 5. we get: Ψ OUT = Ψ OUT C, C = π π From Norlund, formulae. and.3 we get: ϕ i ξ i ξ i Ψ OUT = Ψ OUT C, C =, ; arg λ < π, arg λ < π. π ei π ɛ ; 5

26 < arg λ < π ɛ =, π < arg λ < ɛ =. Note on the computation: In order to apply. we need: g,, ; = g λ,, ; λ + πe i π ɛ F,, ; λ < arg λ < π ɛ =, π < arg λ < ɛ =. ɛ appears in the computations when we epress: λ = e iπɛ λ. We epand the solutions for λ and we get: Ψ ln λ OUT = G + Oλ B, B = Namely, Then epansion when λ yields:: Namely, From the above: It is easy to see that: Ψ OUT = I + O D = Ψ OUT = Ψ OUT B. λ ln 6 r Ψ Match OUT = Ψ OUT D = Ψ OUT C D 4 ln, λ. λ σ3 λ R D, λ ;, R = Ψ OUT = ΨMatch OUT D. r. Ψ OUT C OUT, where C OUT = BC D. Ψ OUT Ψ OUT This, together with the connection formulae πi, when λ e πi λ. Ψ Match OUT = Ψ OUT C D, yields: M M OUT = DC C = Ψ OUT C C D, = C OUT BC πi C C D πi C B C OUT. We have two choices for C OUT, depending on ɛ = ± in C. These have been called C + OUT in the Note, after Proposition 4. C OUT and MATCHING Ψ Ψ IN The system: Φ := µ p Ψ, dφ dµ = Â p + µ Aˆ µ Φ. 6

27 is 7 of Appendi, with: a = p, c = p. The equation for ξ is in Gauss hypergeometic form 77: µµ d ξ dµ + + c a + µ dξ dµ ϕµ = In the standard form 55, we have: aa c µµ dξ + aµ c rξ dµ α = p, β = + p, γ = + p; z = µ. aa + ξ =, 57 Therefore γ = + p, α + β γ + =, α β + =, and 57 may have solutions with logarithmic or polynomial behaviors at µ =,,. The choice of three sets of independent solutions requires a distinction of sub cases p > and p <. As before, we denote: * CASE p >. We choose: { Ψ i = µ p Φ i i ϕ, Φi = ϕ i ξ i ξ i ξ = F α, β, γ ; µ,., i =,,. ξ = µ γ fα γ +, β γ +, γ ; µ; { ξ = F α, β, α + β γ + ; µ, ξ = gα, β, α + β γ + ; µ; { ξ = µ β F β, β γ +, β α + ; µ, ξ = µ β g β γ +, β, β α + ; µ ; From Norlund, formulae 5., 5.7 we get: Ψ = Ψ C From Norlund, formulae.,.6 we get: Ψ = Ψ C where µ = e iπη µ, η = ±., arg λ < π, C =, arg µ < π, C pγp Γp Γp Γp p Γp = p+ Γp p Γp Γp We compute the behavior of ϕ i, ξ i i =, for µ. In the computation, ln /µ appears in g. We write /µ = e iπη /µ, arg µ = ηπ when < µ <. The final result after epanding in series: Ψ ln µ p Q µ = I + O E, µ, E = > p µ p,., Namely, Q > = ψ E p + ψ E p + + γ E + iπη p + r p. Ψ = Ψ E, 7

28 where Ψ is the matri for the matching Ψ OUT Ψ IN. Epanding ϕ i, ξ i for µ we get: r+p Ψ r I = p p + Oµ µ pσ 3, µ. Epanding ϕ i, ξ i for µ we get: Ψ = p+r p p+r p ψ E p + ψ E p + + γ E p ψ E p + ψ E p + + γ E * CASE p <. We choose: { ξ = fα, β, γ ; µ, ξ = µ γ F α γ +, β γ +, γ ; µ; { ξ = F α, β, α + β γ + ; µ, ξ = g α, β, α + β γ + ; µ; { ξ = µ β F β, β γ +, β α + ; µ, ξ = µ β g β, β γ +, β α + ; µ ; From Norlund, formulae 8.6, 8. we compute: Ψ = Ψ C, arg λ < π, C From Norlund, formulae 3., 3.6 we compute: Ψ = Ψ C, arg µ < π, C I ln µ + O µ, µ. pγ p = Γ p p Γ p = p+ Γ p Γ p Γ p. p Γ p Γ p We compute the behavior of ϕ i, ξ i i =, for µ. In the computation, ln /µ appears in g. We write /µ = e iπη /µ, arg µ = ηπ when < µ <. The final result epanding in series: Ψ ln µ p Q µ = I + O E, µ, E = < p µ p, Namely, Q < = ψ E p + ψ E p + + γ E + iπη p + r p. Ψ = Ψ E, where Ψ is the matri for the matching Ψ OUT Ψ IN. Epanding ϕ i, ξ i for µ we get: r+p Ψ r I = p p + Oµ µ pσ 3, µ. Epanding ϕ i, ξ i for µ we get: Ψ = p+r p p+r p ψ E p + ψ E p + γ E p ψ E p + ψ E p + γ E * Both for p > and p < we have: Ψ IN = K Ψ = K Ψ E, I ln µ + O µ, µ. = K Ψ C E, = K Ψ C C E ; 8.

29 together with Ψ Match IN = Ψ IN C OUT. We conclude that the monodromy of is: M M IN = I, M M IN = C OUT EC C πi C C E C OUT. The connection matrices E, C, C have different form for p > and for p <. We also have two choices for C OUT, depending on ɛ = ± in C. These have been called C + OUT and C OUT in the comments just after Proposition 4. Multiplying by C OUT and C OUT to the left and right respectively we get three generators for the monodromy group: M = I, M = BC πi C B, M = EC πi C With this choice, we obtain the matrices of the Proposition 4. We observe that C πi OUT M M M C OUT =, ɛ = ; C OUT M M M C OUT = πi, ɛ =. trm M = trm M =, trm M =. C C E. 8 Logarithmic Behaviors at = and = Symmetries and their Action on the Monodromy Data Connection Problem In this section we compute the logarithmic asymptotic behaviors at =,. This is easily done by applying the action of some Backlund transformations of PVI on 6 and 7. They act as birational transformations on y and, and as permutations on the θ ν s, ν =,,,. In order to know the monodromy data which are associated to the solutions of PVI obtained from 6 and 7 by the Backlund transformations, we also compute their action on the monodromy data. The birational transformations are described in 3; some of them form a representation of the permutation group and are generated by: σ : θ = θ, θ = θ ; θ = θ, θ = θ ; y = y, =. σ : θ = θ, θ = θ + ; θ = θ, θ = θ ; y = y, =. σ 3 : θ = θ, θ = θ ; θ = θ, θ = θ ; y = y, =. It is convenient to consider also: θ = θ, θ = θ ; θ = θ, θ = θ ; y = y, = ; 58 θ = θ, θ = θ, θ = θ, θ = θ + ; y = y, =. 59 θ = θ, θ = θ, θ = θ + ; θ = θ ; y = y y, =. 6 The transformantion 58 is the composition σ σ 3 σ. 59 is σ σ 3. 6 is the composition of σ, 58, 59. For brevity, we will call the Backlund transformations with the name symmetries. 9

30 8. Action on the Transcendent. Formulae -6 and 7- The symmetry σ 3, acting on the transcendent 6, gives the behavior: θ y θ θ + θ θ ln + 4r + θ θ θ, ; We prove below that σ 3 maps trm M to trm M, where M ν, ν =,,, are th emonodromy matrices for the system associated to y, with respect to the same basis of loops see below. Therefore trm M =. The symmetry σ, acting on the transcendent 6, gives the behavior: y θ θ θ + θ θ ln + 4r + θ 4 θ θ,. As it is proved below, σ maps trm M to trm M and thus trm M =. The action of 59 gives the behavior: y θ θ 4 ln + 4r+θ θ θ + θ θ θ,, Namely, y = 4 θ θ ln + 8r + 4θ 4 θ θ ln + O ln,. The symmetry 6 gives: y + θ θ 4 ln + 4r+θ θ θ θ + θ θ,. Namely: y 4 = + θ θ ln 8r + 4θ θ θ ln + O ln, The symmetry σ yields: y θ θ 4,. ln + 4r+θ θ θ θ + θ θ Namely, 4 y = θ θ ln 8r + 4θ θ θ ln + O ln,. We study the action of the symmetries on 7. If we apply σ we find: The action of σ 3 gives: y r ± θ ln,, θ = ±θ. y r ± θ ln,, θ = ±θ. 3