{'fraction_non_alphanumeric': 0.07181523716554998, 'fraction_numerical': 0.029285175489195545, 'mean_word_length': 3.512915498598318, 'pattern_counts': {'":': 0, '<': 77, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Elliptic Representation of the General Painlev\'e 6 Equation

We study the analytic properties and the critical behavior of the elliptic representation of solutions of the Painlev\'e 6 equation. We solve the connection problem for elliptic representation in the generic case and in a non-generic case equivalent to WDVV equations of associativity.


Introduction
In this paper we study the elliptic representation of the sixth Painlevé equation Though the elliptic representation of PVI has been known since R.Fuchs [8], in the literature there is no general study of its analytic implications. To fill this gap, we study here the analytic properties of the solutions in elliptic representation for all values of α, β, γ, δ and we derive their critical behavior close to the singular points x = 0, 1, ∞. Moreover, we solve the connection problem for generic values of α, β, γ, δ and for the special (non-generic) case β = γ = 1 − 2δ = 0, which is important in 2-D topological field theory.
The six classical Painlevé equations were discovered by Painlevé [29] and Gambier [9], who classified all the second order ordinary differential equations of the type where R is rational in dy dx , x and y. The Painlevé equations satisfy the Painlevé property of absence of movable branch points and essential singularities. These singularities will be called critical points; for PVI they are 0,1,∞. The behavior of a solution close to a critical point is called critical behavior. A solution of the sixth Painlevé equation can be analytically continued to a meromorphic function on the universal covering of P 1 \{0, 1, ∞}. For generic values of the integration constants and of the parameters in the equation, it can not be expressed via elementary or classical transcendental functions. For this reason, it is called a Painlevé transcendent.
The Painlevé equations turned out to be important in physical models, since the works [24] [25] [26]. The reader may find a review of the physical applications in [13]. More recently, the Painlevé 6 equation found applications in topological field theory and Frobenius manifolds [4], [5] The first analytical problem with Painlevé equations is to determine the critical behavior of the transcendents at the critical points. Such a behavior must depend on two parameters, which are integration constants. The second problem, called connection problem, is to find the relation between the couples of parameters at different critical points. The method of isomonodromic deformations developed in [16] [17] was applied to the Painlevé 6 equation in [15], to solve such problems for a class of solutions of PVI with generic values of the parameters. The non-generic case β = γ = 1 − 2δ = 0 is studied in [7] [23] [11] for its applications to topological field theory. Studies on the critical behavior can be also found in [35].
In the present paper we show that the elliptic representation is a valuable tool to study the critical behavior of the Painlevé 6 transcendents. We obtain results which include the results of [15] [7] and extend the class of solutions to which they apply. On the other hand, we needed to use the isomonodromic deformation theory to solve the connection problem, to be formulated below, for the elliptic representation.
The elliptic representation was introduced by R. Fuchs in [8]. Let be a linear differential operator and let ℘(z; ω 1 , ω 2 ) be the Weierstrass elliptic function of the independent variable z ∈ P 1 , with half-periods ω 1 , ω 2 . Let us consider the following independent solutions of the hyper-geometric equation Lω = 0: where F 1 2 , 1 2 , 1; x is the standard notation for the hyper-geometric function. R.Fuchs proved that the Painlevé 6 equation is equivalent to the following differential equation for a new function u(x): The connection to Painlevé 6 is given by the following representation of the transcendents: ; ω 1 (x), ω 2 (x) + 1 + x 3 .
We review these facts in section 2.
The algebraic-geometrical properties of the elliptic representations where studied in [22]. Nevertheless, the analytic properties of the function u(x) have not been studied so far, except for some special cases. The most simple case is α = β = γ = 1 − 2δ = 0. The function u(x) is a linear combination of ω 1 and ω 2 . This case was well known to Picard [30], and the critical behavior was studied in [23]. A more general case was studied in [11], for β = γ = 1 − 2δ = 0 and α any complex number. The motivation of [11] was that this case is equivalent to the WDVV equations of associativity in 2-D topological field theory introduced by Witten [36], Dijkgraaf, Verlinde E., Verelinde H. [2] and it has applications to Frobenius manifolds [4] and quantum cohomology [21].
In this paper, we study the analytic properties of u(x) and we compute the critical behavior of the transcendents for any value of α, β, γ, δ; moreover, we solve the connection problem in elliptic representation for generic α, β, γ, δ and for β = γ = 1 − 2δ = 0.
This is a consequence of the periodicity of the ℘-function. Therefore, one can take 0 ≤ ℜν i < 2, i = 1, 2. Nevertheless, we don't need to suppose such a range explicitly. Only in the case ℑν 2 = 0 we need to suppose that 0 ≤ ν 2 < 2. Finally, let us introduce the following expansion: v(x; ν 1 , ν 2 ) := n≥1 a n x n + n≥0,m≥1 b nm x n e −iπν1 x 16 c nm x n e iπν1 x 16 ν2 m (4) Theorem 1: Let ν 1 , ν 2 be two complex numbers.

Critical Behavior
It is possible to compute the critical behavior for x → 0 of a transcendent of Theorem 1. For simplicity, we consider x → 0 along the paths defined below. Let ℑν 2 = 0 and V ∈ C. We define the following family of paths joining a point x 0 ∈ D(r; ν 1 , ν 2 ) to x = 0 The paths are contained in D(r; ν 1 , ν 2 ). If ℑν 2 = 0 any regular path contained in D 0 (r) can be considered.
For V = 1: For ν 2 real we have two cases. For 0 < ν 2 < 1, the transcendent y(x) = ℘(ν 1 ω 1 +ν 2 ω 2 +v(x; ν 1 , ν 2 ); ω 1 , ω 2 )+ (1 + x)/3 defined in D 0 (r) has behavior 2 := ω 1 and ω (∞) 1 in a neighborhood of x = 1, and solutions in a neighborhood of x = ∞. Let P 1 i := P 1 \{i}, i = 1, ∞, and let P 1 i be the universal covering. such that ℑν where it has the convergent expansion: For any complex ν with the constraint 0 < ν 2 < 1, there exists a sufficiently small r and a transcendent y(x) = ℘(ν 2 )) + 1+x 3 [we omit half-periods for semplicity] such that v (1) 2 ) is holomorphic in To summarize the results for the generic case, we first observe that the critical behaviors provided by the elliptic representations along regular paths (except special directions for V = 0, 1, see Theorem 2 and section 6.1) at x = 0, x = 1 and x = ∞ respectively are and the parameters ν (i) 1 are given by 2 is real, the behavior is as above when 0 < ν with Note that the ambiguity ν does not change and the ℘-function is periodic.
Let M 0 , M 1 , M x be the monodromy matrices at z = 0, 1, x, for a given basis in the fundamental group of P 1 \{0, 1, x, ∞}. Such basis is chosen as in figure 2.
there is a one to one correspondence between a given choice of monodromy data θ 0 , θ x , θ 1 , θ ∞ , tr(M 0 M x ), tr(M 0 M 1 ), tr(M 1 M x ) and a transcendent y(x). Namely: We prove that such a transcendent has elliptic representations at x = 0, 1, ∞, provided that (14) is satisfied. The three sets of parameters (ν The formulas of a (i) are quite long, so we do not write them here. They depend on the monodromy data through rational, trigonometric and Γ-functions. In particular, ν (i) 2 enters explicitly. The procedure for computing such formulae is given in the Appendix. We note that the condition ν Conversely, we prove that a transcendent y(x) given by its elliptic representation close to one critical point is a transcendent (21) for some monodromy data. This follows from the consideration that the couple (ν 2 ) is given at the critical point x = i, and θ 0 , θ x , θ 1 , θ ∞ are fixed by the equation PVI we are considering. From these data we can compute tr and Γ-functions. The formulae are rather long, so we refer the reader to the Appendix. In this way the transcendent (21) is obtained. From the monodromy data we compute the couples (ν 2 ) at the other two critical points and we get the the elliptic representation of the initial transcendent at the other critical points. Therefore, the connection problem is solved.
Note that if we start from the elliptic representation at one critical point, say for example x = 0, then ν (0) 1 , ν (0) 2 are given. As explained above, we can compute the monodromy data and from them we compute ν (j) 2 and a (j) (then ν (j) 1 ) at the other two critical points. As already observed, the ambiguity ν (j) 1 → ν (j) 1 + 2k (k integer) does not change the elliptic representation. On the other hand, the ambiguities ν and the ambiguity in the choice 0 ≤ ℜν (j) 2 ≤ 1 or 1 ≤ ℜν (j) 2 ≤ 2, which results from the cosines in (22), is due to the fact that the same transcendent has different elliptic representations in different domains (the choice of ν (j) 2 determines the representation and the domain!).
To conclude the discussion of the generic case, some comments about our extension of previous known results are in order. The critical behavior for a class of solutions to the Painlevé 6 equation was found by Jimbo in [15] for generic values of α, β, γ δ. A transcendent in this class has behavior: where δ is a small positive number, a (i) and σ (i) are complex numbers such that a (i) = 0 and 0 ≤ ℜσ (i) < 1.
We remark that x converges to the critical points inside a sector with vertex on the corresponding critical point. The connection problem, i.e. the problem of finding the relation among the three pairs (σ (i) , a (i) ), i = 0, 1, ∞, was solved in [15] for the above class of transcendents using the isomonodromy deformations theory. Actually, a transcendent in the class above coincides with a transcendent (21). In particular and For the formulas of a (i) we refer to [15]. The monodromy data are restricted by the following condition, equivalent to (26): As explained above, we have shown that the transcendents (21) have elliptic representation. Therefore, Jimbo's transcendents are included in our class of transcendents obtained by the elliptic representation. Observe that the behaviors (23)- (25) are included in the behaviors (15)-(17) (and (18)- (20)), which hold for any ν . Therefore the condition (26) is extended to any σ (i) ∈ C such that σ (i) ∈ (−∞, 0] ∪ [1, +∞). This corresponds to the fact that we have solved the connection problem for any complex value of tr(M i M j ) with the only constraint tr(M i M j ) = ±2. This condition extends (27).
To be more precise, the condition ν Nevertheless, in the case tr(M i M j ) = 2 the critical behavior and the solution of the connection problem were achieved by Jimbo. Unfortunately, the condition ν (i) 2 = 1 which we had to impose to study the elliptic representation (except for non-generic cases like β = γ = 1 − 2δ = 0) does not allow us to know the analytic properties and the critical behavior of the elliptic representation in this case. We expect that the properties of u(x) are such to exactly produce the critical behavior found by Jimbo for tr(M i M j ) = 2, but we still have to cover this case.
The condition ν (i) 2 = 0 (and 2), implies that we can not give the critical behaviors (and the elliptic representation) of (21) To our knowledge, these cases have not yet been studied in the literature.
To conclude, the results of [15] together with our extension provide the critical behaviors and the solution of the connection problem for the transcendents (21) in the generic case for any value of tr(M i M j ) = −2 which corresponds to exponents We turn now to the special case β = γ = 1 − 2δ = 0, important for its applications to topological filed theory, Frobenius manifolds [5] and quantum cohomology [19] [12]. In this case is fully studied in [11]. We can give a representation of u(x) in a domain which is wider than the generic case. Namely, at x = 0, the domain is If ν 2 is real, the value ν 2 = 1 is now allowed, namely, the constraint is ν 2 ∈ (−∞, 0] ∪ [2, +∞). Therefore, by periodicity of the ℘-function we can assume 0 ≤ ℜν 2 < 2, ν 2 = 0. A similar result holds at x = 1 and x = ∞ (see section 8.2). According to [7], we define 2 − There is a one to one correspondence between triples (x 0 , x 1 , x ∞ ) (defined up to the change of two signs) and Painlevé transcendents, provided that at most one x i is zero and not all the x i are ±2 at the same time. Therefore we write y(x) = y(x; x 0 , x 1 , x ∞ ). We show that one such transcendent has elliptic representations (half-periods are understood) The parameters ν (i) 2 are obtained from cos πν is obtained by the formula respectively. The most general choice of ν 2 is 0 ≤ ℜν 2 < 2. This corresponds to the fact that the transcendent y(x; x 0 , x 1 , x ∞ ) also has three representations is obtained by the formula } are given by an analogous formula with the substitutions ( respectively.
Conversely, a transcendent coincides with y(x; x 0 , x 1 , x ∞ ), with the following monodromy data.
We observed that there is a one to one correspondence between Painlevé transcendents and triples of monodromy data (x 0 , x 1 , x ∞ ), defined up to the change of two signs, satisfying x i = ±2, i = 0, 1, ∞, i.e. ν (i) 2 = 0 (and 2), and at most one x i = 0. The cases when these conditions are not satisfied are studied in [23]. However, if x i = ±2 (namely the trace is −2) the problem of finding the critical behavior at the corresponding critical point x = i is still open (except when all the three x i are ±2: in this case there is a one-parameter class of solutions called Chazy solutions in [23]). We conclude that the results of our paper (and [11]), plus the results of [23] cover all the possible transcendents, except the special case when one or two x i are ±2. We plan to cover this last case soon.
Finally, we expect that in all non-generic cases we can solve the connection problem and express the parameters ν 1 , ν 2 in terms of monodromy data. From the conceptual point of view nothing should change with respect to [15] [7] [11] and the present paper; but the technical details may require a long time for computations.

The Elliptic Representation
We derive the elliptic form for the general Painlevé 6 equation. We follow [8]. Let We observe that du dx = ∂u ∂y By direct calculation we have: Therefore, y(x) satisfies PVI if and only if We invert the function u = u(y) by observing that we are dealing with an elliptic integral. Therefore, we write where f (u, x) is an elliptic function of u. This implies that The above equality allows us to rewrite (32) in the following way: where The last step concerns the form of f (u, x). We observe that 4λ(λ − 1)(λ − x) is not in Weierstrass canonical form. We change variable: and we get the Weierstrass form: We still need to explain what are the half periods ω 1 , ω 2 . In order to do that, we first observe that the Weierstrass form is where Therefore g := √ e 1 − e 2 = 1, κ 2 := e 2 − e 3 e 1 − e 3 = x, κ ′ 2 := 1 − κ 2 = 1 − x We identify e 1 = ℘(ω 1 ), e 2 = ℘(ω 1 + ω 2 ), e 3 = ℘(ω 2 ). Therefore, the half-periods are The elliptic integral K(x) coincides, up to a factor, with the hyper-geometric function F 1 2 , 1 2 , 1; x , whose convergent series for |x| < 1 is: Namely: Moreover, K(x) and K(1 − x) are two linearly independent solutions of the hyper-geometric equation Observe that for |arg(x)| < π: Therefore In the following we use sometimes the abbreviation F (x) for F 1 2 , 1 2 , 1; x . The series of F (x) and F 1 (x) converge for |x| < 1.
The PVI equation becomes We recall that We observe also that e 1 − e 3 = 1, e 2 − e 3 = x, e 1 − e 2 = 1 − x. Therefore, (34) becomes: 3 Proof of Theorem 1 For technical reason it is convenient to introduce the following definitions: Note that D 1 (r; ν 1 , ν 2 ) ≡ D(r; ν 1 , ν 2 ) in (2) and D 2 (r; ν 1 , ν 2 ) ≡ D(r; −ν 1 , 2 − ν 2 ). Let be the modular parameter of the elliptic function, obtained expanding ω 1 (x), ω 2 (x) at x = 0. We expand the r.h.s of (35) in Fourier series. We are dealing with Weierstrass ℘-functions of the form Due to periodicity of ℘ we are going to expand As it is well known [33], the Fourier expansion of the ℘-function is We observe that for x → 0 The condition (38) becomes If ν 2 is real, we divide by ℑτ , we let x → 0 and we obtain In the domain defined by the above condition, and |x| < 1, we do the Fourier expansion as follows Note that N 1 does not appear in the expansion. So, we can take N 1 = 0 in the following.
Now we make the further assumption that and we rewrite (41) as follows: Due to (42), the denominator in the first term does not vanish, so that the expansion has no poles. The condition (42) is more restrictive than (40) and yields: We observe that, for any complex number C, In the r.h.s. of equation (35), we have four ℘-functions with different arguments. We recall that we can take N 1 = 0. As for N 2 , it can be different in each ℘-function. There are only two possibilities in order for the four resulting domains (43) to intersect: the ℘-functions must be (35) is expanded in Fourier series if |x| < 1, in the following domains.
Case (a): Case (b): Note that if ℑν 2 = 0 any value of arg(x) is allowed, but dividing the previous expressions by ln |x| and letting x → 0 we have (44) (45) into account in the two cases, we conclude that: In case (a), for N ′ 2 = N 2 = N , the following powers appear in the r.h.s. of (35): The dominant powers are then and we can write (35) as; on the domain defined by |x| < 1 and (46), assuming that v(x) is bounded.
In case (b), for N ′ 2 + 1 = N 2 = N , the following powers appear in the r.h.s. of (35): The dominant powers are then and we can write (35) as; on the domain defined by |x| < 1 and (47), assuming that v(x) is bounded.
Let ǫ < 1 be sufficiently small: in both cases, F (x, y, z) is holomorphic for |x|, |y|, |z| < ǫ and F (0, 0, 0) = 0. In (a) we decompose , v In the same way, in case (b) we obtain a similar system, where , v, w By construction, in both cases Φ(x, y, z) and Ψ(x, y, z, v, w) are holomorphic of their arguments for |x|, |y|, |z|, |v|, |w| < ǫ for sufficiently small ǫ < 1 (assuming v(x) bounded in a neighborhood of x = 0). Moreover We complete the proof of the theorem in case (a). Case (b) is analogous. We reduce the system of differential equations to a system of integral equations The point x and the path of integration are chosen to belong to the domain where |x|, | e −iπν 1 x ν2+2N |, |v(x)|, |w(x)| are less than ǫ, in such a way that Φ and Ψ are holomorphic. That such a domain is not empty will be shown below. In particular, we'll show that if we require that |x| < r, where r < ǫ is small enough, also the solutions of the integral equations |v(x)| and |w(x)| are less than ǫ. Such a domain is contained in (46).
We choose the path of integration L(x) connecting 0 to x, defined by If x belongs to the domain (46) in case (a) and to (47) in case (b), than the path does not leave the domain when s → 0, provided that 0 ≤ ν * ≤ 1 in case (a), or 1 ≤ ν * ≤ 2 in case (b). If ℑν 2 = 0 we take the radial path arg s = arg x, 0 < |s| ≤ |x|, namely ν * = ν 2 . The parameterization of the path is For any complex numbers A, B we have from which (48) follows, provided that 0 < ν * < 1. For ℑν 2 = 0, this yields again 0 < ν 2 < 1.
We observe that a solution of the integral equations is also a solution of the differential equations, by virtue of the following lemma: Proof: We give the proof in case (a). We choose a point x + ∆x close to x and we prove that , where the last integral is on a segment from x to x+ ∆x. We consider a small disk U R centered at x = 0 of small radius R < |x| and the points x R := L(x)∩U R , x ′ R := L(x+∆x)∩U R . Since the integral of f /s on a finite close curve (not containing 0) is zero we have: The last integral is on the arc γ(x R , x ′ R ) from x R to x ′ R on the circle |s| = R. We have also kept into account the obvious fact that L(x R ) is contained in L(x) and L(x ′ R ) is contained in L(x + ∆x). We then take R → 0 and we prove that the r.h.s. of (49) vanishes.
Taking into account that The last step follows from |x ν2+2N So the integral vanishes for R → 0. The same is proved for L(x+∆x) . As for the integral on the arc we have Now we prove a fundamental lemma.
Moreover, for δ small: which implies that ∂Ψ ∂x is holomorphic and bounded when its arguments are less than ǫ. The same holds true for ∂Ψ ∂y and ∂Ψ ∂z . This proves (51), c being a constant which bounds ∂Ψ ∂x , ∂Ψ ∂y ∂Ψ ∂z . The inequality (50) is proved in the same way. We turn to (52). First we prove that for |x|, |y|, |z|, |v 1 |, |w 1 |, |v 2 |, |w 2 | < ǫ there exist two holomorphic and bounded functions In order to prove this, we write Moreover, for small δ, which implies that ψ 1 is holomorphic and bounded for its arguments less than ǫ. We also obtain The proof for ψ 2 is analogous. We use (53) to complete the proof of (52). Actually, we observe that and we conclude as in the proof of (51). ✷ We solve the system of integral equations by successive approximations. We can choose any path L(x) such that 0 < ν * < 1 in case (a), or 1 < ν * < 2 in case (b). Here we complete the proof only for the case (a). Therefore, the solution v(x) we are going to find is v 1 (x; ν 1 , ν 2 + 2N ). Case (b) is analogous and we don't need to repeat the proof for it. It yields the function v 2 (x; ν 1 , ν 2 + 2N ).
It follows from Lemma 1 and (50) that w 1 (x) is holomorphic for |x|, |Ax 1−ν2−2N |, |Bx ν2+2N | < ǫ. From (48) and (50) we have By (48) we also have (57) (56) are true for n = 1. Now we suppose that the statement of the sub-lemma is true for n and we prove it for n + 1. Consider: This proves (57). Now we estimate This proves (56). From Lemma 1 we also conclude that w n and v n are holomorphic in D(ǫ ′ , ν 1 , ν 2 ) and x dv n dx = w n Finally we see that We see that the series converges uniformly in D(ǫ ′ , ν 1 , ν 2 + 2N ) because The same holds for w n (x). Therefore, v(x) and w(x) define holomorphic functions in D 1 (ǫ ′ , ν 1 , ν 2 + 2N ). From Sub-Lemma 2 we also have We show that v(x), w(x) solve the initial integral equations. The l.h.s. of (54) converges to w(x) for n → ∞. Let's prove that the r.h.s. also converges to We have to evaluate the following difference: The series converges. Its sum is less than some constant S(ν 2 + 2N ) independent of n. We obtain The same holds for |w − w n |. Thus, (58) is Namely: In a similar way, the r.h.s. of (55) is Therefore, the r.h. sides of (54) (55) converge on the domain D(r, ν 1 , ν 2 ) for r < min{ǫ ′ , 1/3}. We note that r = r N , namely it depends on N in the same way as ǫ ′ does. Thus, it decreases as N −2 as N increases.
We finally observe that |v(x)| and |w(x)| are bounded on D(r). For example where the sum of the series is less than a constant M (ν 2 + 2N ). We have proved Lemma 2 for the case (a). Case (b) is analogous. ✷ We invite the reader to observe the ln r N terms. Also, note that for ν 2 ∈ R the domain is specified by max |x|, e −iπν 1 16 1−ν 2 x 1−ν2 , e iπν 1 16 ν 2 x ν2 < r. We can make the notation easer by simply re-writing |x| < r for sufficiently small r. So, we obtain the domain D 0 (r).
We have proved that the structure of the integral equations implies that v(x) is bounded (namely |v(x)| = O(r N )). The proof of Lemma 2 only makes use of the properties of Φ and Ψ, regardless of how these functions have been constructed.
To conclude the proof of Theorem 1, we have to work out the series of v(x). Let's do that in case (a), namely for v 1 (x; ν 1 , ν 2 + 2N ). Case (b) is analogous. We observe that w 1 and v 1 are series of the type p,q,r≥0 where c pqr (ν 2 + 2N ) is rational in ν 2 . This follows from and from the fact that Φ(x, Ax 1−ν2−2N , Bx ν2+2N ) itself is a series (59) by construction, with coefficients which are rational functions of ν 2 + 2N . The same holds true for Ψ. We conclude that w n (x) and v n (x) have the form (59) for any n. This implies that the limit v(x) is also a series of type (59). We can reorder such a series. Consider the term and recall that by definition B = 1 16 2 A . We absorb 16 −2r into c pqr (ν 2 ) and we study the factor We have three cases: 1) r = q, then we have x p+q =: x n , n = p + q.
2) r > q, then we have x p+q 1 A x ν2+2N r−q =: x n 1 A x ν2+2N m , n = p + q, m = q − r. 3) r < q, then we have A q−r x p+r Ax 1−ν2−2N q−r =: x n Ax 1−ν2−2N m , n = p + r, m = q − r. Therefore, the series of the type (59) can be re-written in the form of v(x) of Theorem 1. The series of v(x) is uniquely determined by the Painlevé equation, because it is constructed from Φ and Ψ by successive approximations of the solutions of the integral equations.
This concludes the proof. We only remark that if ν 2 is real, we make a translation ν 2 → ν 2 + 2N which yields 0 < ν 2 < 1 or 1 < ν 2 < 2 and all the formulae can be read for N = 0.
The statement of Theorem 1 as it follows from our proof is Let ν 1 , ν 2 be two complex numbers. I) For any N ∈ Z and for any complex ν 1 , ν 2 such that there exist a positive number r N < 1 and a transcendent There also exists a transcendent For both transcendents, the coefficients a (i) II) For any complex ν 1 and for any real ν 2 such that 0 < ν 2 < 1 or 1 < ν 2 < 2 there exists a positive r < 1 and a transcendent such that v(x; ν 1 , ν 2 ) is holomorphic in D 0 (r; ν 1 , ν 2 ), where it has the convergent expansion v(x; ν 1 , ν 2 ) = n≥1 a n x n + n≥0,m≥1 b nm x n e −iπν1 x 16 The coefficients are rational functions of ν 2 and |v(x)| is bounded as in (60), with N = 0, for 0 < ν 2 < 1 and as in (61), with N = 0, for 1 < ν 2 < 2.
The above statement is completely equivalent to the statement of Theorem 1 in the Introduction, due to Observation 1 and 2 there.
A technique similar to that used in the proof of Theorem 1 was first introduced by S. Shimomura in [35] for a class of functional equations.

4
Proof of Theorem 2 Theorem 2 is stated in the Introduction. Here we give its proof. Let x 0 ∈ D(r; ν 1 , ν 2 ). We expand y(x) in Fourier series. This is possible on D(r; ν 1 , ν 2 ) as it is clear from the proof of Theorem 1. Let Observe that for x → 0 π 2 Thus The above formula immediately yields (7) and (10), (11). It is enough to recall that in D(r; ν 1 , ν 2 ) the dominant powers are x ν2 , x 1−ν2 . We also need to observe that and that v(x) → 0 for x → 0, according to (5). Therefore, (7) and (10), (11) follow simply taking the leading terms in the expansion. As for (9), we observe that if V = 1, v(x) does not vanish, namely: ). This implies that the dominant terms in the Fourier expansion are: from which (9) follows. Note that x, x ν2 and x 2−ν2 are of the same order.

Special Cases
There are some special cases when the domain D(r; ν 1 , ν 2 ) can be enlarged.
Theorem 1 can be stated as follows: Theorem 1 -special case β = 1 − 2δ = 0 : For any complex ν 1 , ν 2 with the constraint there exist a positive number r < 1 and a transcendent It has convergent expansion: v(x; ν 1 , ν 2 ) = n≥1 a n x n + n≥0,m≥1 b nm x n e −iπν1 x 16 Note that the domain is larger than the general case and that ν 2 = 1 is now allowed.
For 0 < V < 1: For −1 < V < 0: For V = 1: For V = −1: For V = 0: For ν 2 real, the behaviors on D 0 (r) are: 6 Points x = 1, ∞ -Comments to Theorem 3 and Critical behavior Two independent solutions of the hyper-geometric equation L(u) = 0 are the hyper-geometric function F (x; 1/2, 1/2, 1) and The following connection formulae hold [27]: We recall that ω 1 (x) and ω 2 (x) are π/2 F (1/2, 1/2, 1; x) and −i/2 g(1/2, 1/2, 1; x) respectively. This representation is convenient in a neighborhood of x = 0. In this section we will use the notation ω Taking into account the above connection formulae, we define in a neighborhood of x = 1: We also define, in a neighborhood of x = ∞: The above functions have branch cuts specified by the constrains on arg x. Once they are so defined, they are continued on the universal covering of P 1 \{0, 1, ∞}.
The same procedure is applied to Now we denote by D(r; ν 2 ) the domain around x = 1. The effect of the loop 2 ) if also x ∈ D(r; ν 2 ).
The procedure is applied to , ω provided that also x ∈ D(r; ν ).

Connection Problem
The analysis so far developed is only local. Two natural questions arise.
Question 1). This question makes sense in ν 2 is not real. Suppose that y(x) has the representation of Theorem 1 close to x = 0: If this happens, we should be able to express ν ′ 1 as a function of ν 1 , ν 2 , N . )?
The answer to this questions is positive, but it can not be achieved with the method so far used, which only allows to obtain local results. For example, suppose we want to solve question 2) for y(x) represented as in Theorem 1. We can write There also exists a transcendent where v (1) (x) is represented by Theorem 3. In particular, v (1) (x) is bounded as x → 1. On the other hand, the function v (0) is only known for small x, and it may diverge as x → 1. In the best hypothesis, we may suppose that after the rescaling vanishes as x → 1 (this may happen, since the divergence of ω (1) 2 , which is ln(1 − x) as x → 1, may cancel the divergence of v (0) (x) as x → 1). Therefore, (84) is a transcendent (85) where ν Unfortunately, we are not able to say if the above rescaling is possible using the local analysis. And if it is, we do not know δν 1 , δν 2 .
The answer to questions 1) and 2) can be obtained by the method of isomonodromic deformations (which, on the other hand, has some limitations in providing the local behavior).

Picard Solutions
Before answering Questions 1 and 2 in general, we recall that there is one case when the local analysis in elliptic representation becomes global, namely when v(x) = 0. This happens when α = β = γ = 1 − 2δ = 0. This case was already well known to Picard [30] and it was studied in [23]. The equation (35) reduces to the hyper-geometric equation L(u) = 0, and it has general solution u(x)/2 = ν 1 ω Question 1) can be answered immediately. There is no need to prove Theorem 1 because we have no 2 (x) in the argument of ℘ is defined for any x = 0, 1, ∞. We obtain the critical behavior by Fourier-expanding For any fixed N , the expansion is performed if |x| < 1 and Namely: Note that this defines a domain D(r; ν 1 , ν 2 + 2N ) which contains the union of D 1 (r < 1; ν 1 , ν 2 + 2N ), D 2 (r < 1; ν 1 , ν 2 + 2N ), D 1 (r < 1; ν 1 , ν 2 + 2[N + 1]), D 2 (r < 1; ν 1 , ν 2 + 2[N + 1]) introduced in the proof of Theorem 1. The critical behavior along is: We observe that the choice of N is arbitrary, therefore the same transcendent has different critical behaviors on different domains (86) specified by different values of N . This answers Question 1.
Remark: Note that in the cases V = −2, 0, 2, the sin 2 (...) may vanish, therefore there may be (movable) poles. Actually, this happens because the domain now is bigger than that of the generic case since we did not impose that sin 2 (...) = 0, as we did in the proof of Theorem 1. This is also the reason why we have to keep F1(x) F (x) in the argument of sin 2 (...) in the Fourier expansion.

Solution of the Connection Problem in a Non-Generic Case
We first give the solutions of Questions 1) and 2) in the special case β = γ = 1 − 2δ = 0 studied in [7] and in [11], because of its connection to 2-D topological field theory and to Frobenius Manifolds [4] [5] and quantum cohomology [19] [21] [12]. This sub-section mainly reviews the results of [11] and connects them to the framework of the present paper. In the next sub-section we will consider the generic case. The Painlevé VI equation is the isomonodromic deformation equation of the fuchsian system [17] dY dz where A i (x) (i = 0, 1, x) are 2 × 2 matrices. They depend on x in such a way that the monodromy matrices at z = 0, 1, x, do not change for small deformations of x. Moreover, The constants θ ν , ν = 0, 1, x, ∞ are linked to the parameters of PVI: , where the function κ(x) does not interests us now (see [17]). The monodromy matrices give a representation of the fundamental group of P 1 \{0, 1, x, ∞}. They are usually denoted by M ν , ν = 0, 1, x, ∞ and they correspond to counterclockwise loops around z = 0, 1, x, ∞ respectively. We order the loops as in figure 2 (an arbitrary base point is chosen). The monodromy matrices are not independent, because where b is a complex number. Following [7], let α =: (2µ−1) 2 2 (namely θ ∞ = 2µ, θ 0 = θ x = θ 1 = 0) and: They satisfy For any fixed µ, it is proved in [7] that there is a one to one correspondence between Painlevé transcendents and triples of monodromy data (x 0 , x 1 , x ∞ ), defined up to the change of two signs, satisfying x i = ±2, i = 0, 1, ∞ and at most one x i = 0. A transcendent in one to one correspondence to (x 0 , x 1 , x ∞ ) will be denoted y = y(x; x 0 , x 1 , x ∞ ).
II) For x 2 0 = 4 sin 2 (πµ) (and then x 2 ∞ = −x 2 1 exp(±2πiµ) as it follows from (89)), there are four sub-cases: Conversely, let be given a transcendent with behavior y(x) = ax 1−σ (1 + higher orders) as x → 0, for given σ and a. We define a triple (x 0 , x 1 , x ∞ ) by the formulae below: x 0 = 2 sin π 2 σ Again, there are limiting cases: x 0 = 2 sin(πµ) In all the above formulae the sign of the square roots is arbitrary (a triple is define up to the change of two signs). The relation It is proved in [11] that if the triple (x 0 , x 1 , x ∞ ) defined above is such that x i = ±2 (i = 0, 1, ∞) and at most one x i = 0, then there exists only one transcendent with behavior y(x) = ax 1−σ (1 + higher orders) as x → 0 specified by the given a and σ. It coincides with y(x; x 0 , x 1 , x ∞ ).
Let us return to the elliptic representation. We observe that in the special case β = γ = 1 − 2δ = 0 there is only ∂ ∂u ℘(u/2) in the r.h.s. of (35). Therefore, ε 1 = ε 2 = 0 in the proof of Theorems 1 and 3. As a consequence, not only at x = 0, but also at x = 1, ∞ the domain D is larger. More precisely, at x = 0 the result of section 5 applies: For any N ∈ Z, and for any complex ν m At x = 1. For any N ∈ Z, and for any complex ν For any N ∈ Z, and for any complex ν is real, there exists a transcendent y = ℘ ν where it has a convergent expansion Note that we have used the same notations for the coefficients a n , b nm , c nm both at x = 0, 1 and x = ∞. This is for simplicity of notations, but they are different! We are ready to identify the transcendents y(x; x 0 , x 1 , x ∞ ) with the transcendents in elliptic representation. We know that for x → 0 the transcendent y(x) = ℘ ν 1 ω On the other hand, a transcendent y(x; x 0 , x 1 , x ∞ ) has behavior (91): y(x; x 0 , x 1 , x ∞ ) = ax 1−[±σ+2n] (1 + higher orders) As we mentioned above, the critical behavior uniquely determines the transcendent. Therefore, the transcendent in the elliptic representation, for given ν 1 , ν 2 , N , must coincide with some y(x; x 0 , x 1 , x ∞ ) having the same critical behavior. Let us choose the convention 0 ≤ ℜσ ≤ 1, 0 ≤ ℜν 2 < 2 We take N = n = 0. A necessary condition for the transcendents to coincide is that D(ǫ 0 ; σ) ∩ D(r 0 ; ν 1 , ν 2 ) = ∅ They actually coincide when they have the same critical behavior, namely if: (recall that σ = 1 and ν 2 = 0, 2). Equivalently: We are now ready to answer Question 1) and Question 2). Actually, this is possible because the identifications 1) and 2) above allow us to express the parameters ν 1 , ν 2 in terms of the triple of monodromy data (x 0 , x 1 , x ∞ ) and µ.
Answer to Question 2). In [7], [11] the connection problem was solved by showing that a given transcendent y(x; x 0 , x 1 , x ∞ ) has critical behaviors: where cos(πσ (i) ) = 1 − The parameters a (0) , σ (0) stand for a, σ used before. a (1) , a (∞) are obtained by the formula (92) (or limit cases) with the substitutions (x 0 , respectively. The behavior (97) holds in a domain where M > 0 is sufficiently big; the behavior (96) holds in At x = 1, x = ∞ we repeat the analysis we did above for x = 0, identifying y(x; x 0 , x 1 , x ∞ ) with a transcendent in elliptic representation with the same critical behaviors (by uniqueness of the critical behavior). Thus, we conclude that the transcendent y(x; x 0 , x 1 , x ∞ ) has a representation at x = 1 and it has a representation where the parameters ν (i) 2 are obtained from cos πν The limit cases are left as an exercise for the reader). After computing the monodromy data, we can write the elliptic representations of y(x; x 0 , x 1 , x ∞ ) at x = 1 and x = ∞. These are the elliptic representations at x = 1, x = ∞ of (100). Thus, we have solved the connection problem for (100).
We observed that there is a one to one correspondence between Painlevé transcendents and triples of monodromy data (x 0 , x 1 , x ∞ ), defined up to the change of two signs, satisfying x i = ±2, i = 0, 1, ∞ (i.e. σ (i) = 1) and at most one x i = 0. The cases when these conditions are not satisfied are studied in [23]. However, if x i = ±2 (namely the trace is −2) the problem of finding the critical behavior at the corresponding critical point x = i is still open (except when all the three x i are ±2: in this case there is a one-parameter class of solutions called Chazy solutions in [23]). We conclude that the results of our paper (together with [11]), plus the results of [23] cover all the possible transcendents, except the special case when one or two x i are ±2. We plan to cover this last case soon.

Solution of the Connection Problem in the Generic Case
We solve the connection problem for the elliptic representation in the generic case: We show in the Appendix, following [15], that if θ 0 , θ x , θ 1 , θ ∞ ∈ Z there is a one to one correspondence between transcendents and the monodromy data θ 0 , θ In the Appendix it is shown that for any value of α, β, γ, δ, for any a = 0, for any complex σ ∈ (−∞, 0] ∪ [1, +∞) and for any additional real parameters 0 <σ < 1, ϑ 1 ,ϑ 2 there exists a sufficiently small ǫ and a transcendent y(x; σ, a) in the domain D(ǫ; σ) defined by (90). It has critical behavior y(x; σ, a) = ax 1−σ (1 + higher orders) for x → 0 along a regular path in D(ǫ; σ).
In the Appendix it is also proved that in the generic case a transcendent (101) associated to the monodromy data θ 0 ,θ x ,θ 1 ,θ ∞ , tr(M 0 M x ),tr(M 0 M 1 ), tr(M 1 M x ), coincides with a transcendent y(x; σ, a) when x lies in the domainD(ǫ; σ). σ and a are obtained from the monodromy data by explicit formulae The formula for a is rather long and complicated, and we do not need its explicit form here; therefore we refer to the Appendix for its computation. Note that σ is defined up to σ → ±σ + 2n, n integer. We can fix 0 ≤ ℜσ ≤ 1; thus, a transcendent (101) associated to the monodromy data coincides with the transcendents y x; ±σ + 2n, a ±σ + 2n; θ 0 , θ in the domains D(ǫ n ; ±σ + 2n) defined for sufficiently small ǫ n (note that only 0 < σ < 1 is allowed if σ is real, and in this case no translation ±σ + 2n is possible). We observe that a depends on the monodromy data, but also explicitly on σ. Namely, it changes if we do σ → ±σ + 2n (being the monodromy data fixed). In the following, for brevity we write a = a(σ), understanding the dependence on the monodromy data.
We finally remark that in the Appendix the formula a = a σ; θ 0 , θ [15] also the case σ = 0 is calculated. On the other hand the cases ℜσ < 0 and ℜσ ≥ 1 are not considered in [15], so we have included them in the Appendix.
The claim is proved in Proposition A1 in the Appendix. As a consequence, a transcendents y(x; σ, a) coincides with a transcendents in elliptic representation with the same critical behavior.
In the generic case, we prove in the Appendix the one-to-one correspondence between transcendents and monodromy data. However, the condition σ (i) = 1, namely ν In the generic case we have also assumed σ ∈ Z; nevertheless, in the case σ (i) = 0 (tr(M i M j ) = 2) the critical behavior and the solution of the connection problem can be found in the paper [15] by Jimbo. We note that the corresponding ν (i) 2 should be equal to 1, but unfortunately the condition ν (i) 2 = 1 which we had to impose to study the elliptic representation (except for special cases like β = γ = 1 − 2δ = 0) did not allow us to know the analytic properties and the critical behavior of the elliptic representation in this case. We expect that the properties of u(x) are such to exactly produce the critical behavior found by Jimbo for σ (i) = 0, but we still have to cover this case.

APPENDIX
We give a brief account of the solution of the connection problem for the generic PVI, following [15], with the extension of the values of monodromy data for which the results apply. In [15] the case |tr(M i M j )| > 2 is not considered, so we have to do it now. This extension is necessary to identify the transcendent with the elliptic representation. Such a generalization is proved exactly as in [11], to which we refer for a detailed analysis of the non-generic case β = γ = 1 − 2δ = 0 (already reviewed in section 8.2).
The Painlevé VI equation is the isomonodromic deformation equation of the fuchsian system (87). The Fuchsian system is obtained from the following Riemann-Hilbert problem. We fix θ 0 , θ x , θ 1 , θ ∞ ∈ Z and monodromy matrices M 0 , M x , M 1 corresponding to the loops in the basis of figure 2, with eigenvalues exp{±iπθ i }, i = 0, x, 1 respectively. At infinity, the monodromy is M 1 M x M 0 , and we require that this has eigenvalues exp{±iπθ ∞ }. Having assigned the monodromy matrices and their eigenvalues, there exist 2 × 2 invertible matrices C 0 , C x , C 1 , C ∞ such that In order to construct the fuchsian system, we have to find a 2×2 invertible matrix Y (z; x) holomorphic in P 1 \{0, x, 1, ∞} with the assigned monodromy at z = 0, 1, x, ∞. We also fix the indices of the Riemann-Hilbert problem, namely we require that Y (z; x) be normalized as follows: Here G i are invertible matrices. The fuchsian system is obtained from The fuchsian system satisfies (88). We remark that we have to assume θ 0 , θ x , θ 1 , θ ∞ ∈ Z, otherwise the general form of Y (z; x) would be different from (114).
A 2 × 2 Riemann-Hilbert always has solution [1]. In our case, the solution is unique, up to diagonal conjugation A i → DA i D −1 , where D is any diagonal matrix. To prove this fact, let C ν ,C ν , ν = 0, x, 1, ∞, be such that If follows that C νC −1 ν is any diagonal matrix. We denote Y andỸ the solutions (114) with C ν and C ν respectively. Since C νC −1 ν is diagonal we conclude that YỸ −1 is holomorphic at z = 0, 1, x, ∞. Therefore, being holomorphic on P 1 , it is a constant diagonal matrix D := C ∞C −1 ∞ . This proves uniqueness up to diagonal conjugation. We remark that if some θ ν (ν = 0, x, 1, ∞) is integer, the uniqueness may fail. The cases when this happens if β = γ = 1 − 2δ = 0 are discussed in [23].
Once the Riemann-Hilbert problem is solved, we compute y(x) from A(z; x) 12 = 0. It is clear that y(x) depends on the monodromy data M 0 , M x , M 1 , θ ν (ν = 0, 1, x, ∞). Note that for any invertible matrix C we obtain the fuchsian system (115) from any Y (z; u)C. The solution Y (z; u)C corresponds to the monodromy matrices C −1 M i C. This implies that y(x) depends on the invariants of M i (i = 0, x, 1) with respect to the conjugation M i → C −1 M i C. They are traces and determinants of the products on the M i 's. The determinants are 1, the traces of M i are specified by the eigenvalues and the traces of the products M i1 M i2 ...M in for n > 2 are functions of the traces of M i1 M i2 . Hence: PVI is equivalent to the Schlesinger equations which ensure the isomonodromicity of the fuchsian system [16]: The system (116) is a particular case of where the functions f µν , g µν , h µν are meromorphic with poles at x = 1, ∞ and Let σ be a complex number such that σ ∈ (−∞, 0) ∪ [1, +∞) and let us consider the domain (90) with additional parameters ϑ 1 , ϑ 2 ∈ R, 0 <σ < 1. In [11] the reader can find the proof of the following lemma, which is a generalization in D(ǫ; σ) of [31], page 262 Lemma A1: Consider matrices B 0 ν (ν = 1, .., n 2 ), A 0 µ (µ = 1, .., n 1 ) and Λ, independent of x and such that .
For any 0 <σ < 1 and ϑ 1 , ϑ 2 real there exists a sufficiently small 0 < ǫ < ǫ ′ such that the system (117) has holomorphic solutions A µ (x), B ν (x) in D(ǫ; σ) satisfying: Here C is a positive constant andσ < σ 1 < 1 Lemma A2: Let θ ∞ = 0. Let r, s be two complex numbers not equal to zero. LetĜ 0 be such that The general solution of The connection matricesĈ 0 ,C 0 ,C 1 can be computed explicitly because the 2 × 2 fuchsian systems (123) (124) can be reduced to the hyper-geometric equation. is equivalent to the system where Ψ = y (z − 1) dy dz .
Proof: By direct computation. ✷ Note that the form of G(z) ensures that if y 1 , y 2 are independent solutions of the hyper-geometric equation, then a fundamental matrix of (133) may be chosen to be X(z) = y 1 (z) y 2 (z) * * .