A Riemann-Hilbert Approach to the Heun Equation

We describe the close connection between the linear system for the sixth Painlev\'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.


Introduction
General Heun equation (GHE) [14] is the 2nd order linear ODE with four distinct Fuchsian singularities depending on 6 arbitrary complex parameters. Without loss of generality, three of the singular points can be placed at 0, 1 and ∞ while the position a of the fourth singularity remains not fixed. The canonical form of the general Heun equation (GHE) reads The GHE is the classical example of the Fuchsian ODE that does not admit any continuous isomonodromy deformation. To overcome the difficulty in the construction of the isomonodromy problem, R. Fuchs [7,8] added to four conventional Fuchsian singularities one apparent singularity that is presented in the equation but is absent in the solution. Position y of this apparent singularity is changed together with the position of the Fuchsian singularity x according to the second order nonlinear ODE known now as the sixth Painlevé equation P VI . In [5], it was observed that under certain assumptions the apparent singularity disappears at the critical values and movable poles of y, and the linear Fuchsian ODE turns into the general Heun equation.
In the present paper, instead of the scalar second order differential equation with apparent fifth singular point we shall use its first order 2 × 2 matrix version with four Fuchsian singular points. We will give a detailed proof that the general Heun equation appears at the poles of P VI , formulate the Riemann-Hilbert (RH) problem for the general Heun equation and explore some implications of the RH problem scheme to the Heun transcendents. In fact, we show how one can obtain the Heun polynomials (i.e., polynomial solutions of (1.1)) within the suggested Riemann-Hilbert formalism.
2 Reduction of the linear differential system for P VI to the general Heun equation (GHE)

Isomonodromy deformations of a Fuchsian linear ODE with four singularities
The modern theory of the isomonodromy deformation was developed in the pioneering work of M. Jimbo, T. Miwa, and K. Ueno [18,19], although its origin goes back to the classical papers of R. Fuchs [7], R. Garnier [9,10], and L. Schlesinger [24]. We shall briefly outline the theory in the case of the 2 × 2 matrix Fuchsian ODE with four singular points. The reader can find more details in the works [18,19] (see also Part 1 of the monograph [5]). The generic first order 2 × 2 matrix Fuchsian ODE with four singular points can be written in the form dΨ dλ = A(λ)Ψ, (2.1) with the coefficient matrix where σ 3 = 1 0 0 −1 . Below we assume that Tr A(λ) ≡ 0, and denote by ±α j , j = 1, 2, 3, 2α j / ∈ Z the eigenvalues of the matrix residues A j . The deformation, with respect to the position of the singularity λ = x is isomonodromic iff Ψ(λ) ≡ Ψ(λ, x) satisfies an auxiliary linear ODE with respect to this variable In [19], it is shown that the unique zero y ≡ y(x) of the entry A 12 (λ)( which is a rational function whose numerator is linear in λ) satisfies the classical sixth Painlevé equation P VI , Suitable parameterization of the coefficient matrix A(λ) of the Fuchsian equation (2.2) and appearance of the sixth Painlevé equation P VI satisfied by this zero y(x) are explained in more detail in Appendix A.1.
In [5, p. 86], it was observed that, at the critical values y = 0, 1, x and movable poles y = ∞, the linear matrix equation (2.2) with the parametrization (A.1) becomes equivalent to the GHE. In the following subsections, we describe the way in which the Heun equation emerges at the movable poles of the Painlevé functions with more details than in [5].

Movable poles of P VI
If δ = 1 2 , then equation P VI admits a 2-parameter family of solutions with the following leading terms [11] of the Laurent expansion where a ∈ C\{0, 1}, c 0 ∈ C are arbitrary, while all other coefficients are determined recursively by a, c 0 , σ = ±1 and the local monodromies α j , j = 1, 2, 3,

etc.
2.3 The coefficient matrix A(λ) at the movable poles of P VI In this subsection, our concern is the behavior of A(λ) at the poles of y(x). We shall use the parameterization of A(λ) given in (A.1). As we will see, the coefficient matrix is continuous at the simple poles with positive σ (that is, δ = 1 2 , σ = +1) and is singular at the poles of any other kind. In the latter case, the linear ODE can be regularized by a suitable Schlesinger transformation [19], and all three resulting regular linear ODEs are equivalent to the GHE.
Theorem 2.1. At any pole of a solution to P VI , the associated linear ODE (2.1) is equivalent (in some cases after a suitable regularization) to GHE (1.1). Moreover, the pole position becomes the position of the fourth singularity in GHE while the free parameter of the Laurent expansion of the Painlevé function determines the accessory parameter in GHE.
Remark 2.2. The part of this statement concerning the relation of the free parameter of the Laurent expansion of the Painlevé function and the accessory parameter in GHE, in the case of δ = 1 2 , has been already established in [20]. In [20] the authors are using a very different approach based on the discovered in [20] remarkable connection of the classical conformal blocks and the sixth Painlevé equation. The authors of [20] also make use of the striking fact (first observed in [26]) that the Heun equation can be thought of as the quantization of the classical Hamiltonian of P VI .
In Sections 2.3.1-2.3.5, we give the detailed proof of Theorem 2.1 considering each case individually.
2.3.1 δ = 0, 1 2 , and σ = +1 (regular case) In this case, the coefficient matrix remains continuous. In more details, using (2.3) along with (A.2), one finds This zero cancels the simple pole of y(x), so the (1, 2)-entry of the matrix A(λ) remains bounded. Furthermore, the direct computer-aided computation yields the continuity of all other entries as well, Here and below σ + = ( 0 1 0 0 ), σ − = ( 0 0 1 0 ). The coefficients a 3 , b 3 , c 3 , c + , b − , c − are expressed in terms of the local monodromies δ, α 1 , α 2 , α 3 , the position of the pole a and the coefficient c 0 of the Laurent series (2.3). Complete details can be found in Appendix A.2.
2.3.2 δ = 0, 1 2 , 1 and σ = −1 (generic singular case) Now, the function κ develops a pole as x → a, that regularizes the equation (2.2) as x → a. Assume first that δ = 1. The needed Schlesinger transformation shifts the formal monodromy at infinity δ by −1, i.e., δ → δ − 1; it is given explicitly by It is straightforward to check that the transformed matrixÂ remains regular at the pole x = a, and the limiting coefficient matrix is as followŝ where expressions for the constant parametersâ 3 ,b 3 ,ĉ 3 ,b + ,ĉ + ,ĉ − in terms of the parameters a, c 0 and the local monodromies can be found in Appendix A.3.

2.
3.3 δ = 1, σ = −1 (the first special singular case) Let us proceed to the case δ = 1 and σ = −1. We choose the Schlesinger transformation that changes the formal monodromy at infinity and at the origin by one half and is given by the gauge matrix R 1 (λ), The transformed coefficient matrixǍ is regular at the pole x = a, and its limiting value is as followsǍ where the explicit expressions for the constant coefficients are given in Appendix A.4. If δ = 1 2 then the Laurent expansion with the double pole (2.4) implies that the coefficient matrix A(λ) is singular at x = a. The chosen regularizing Schlesinger transformation shifts δ → δ + 1, The transformed matrix,Ã = R 2 AR −1 2 + R 2λ R −1 2 , is regular at the pole x = a and, at this point, takes the valuẽ

GHE from the linear ODEs at the poles of P VI
In this subsection, we show that all the linear matrix ODEs corresponding to the poles of the sixth Painlevé function are equivalent to the GHE.
Consider first the coefficient matrix (2.5). The first order matrix equation for the function Ψ(λ) is always equivalent to the second order Fuchsian ODE for the entry Ψ 1 * (λ) of the first row of the matrix function Ψ(λ). However, extra (apparent) singularities might appear in the process of excluding the entry Ψ 2 * (λ). In the case of (2.5), however, the rational function representing the 12 -entry of matrix (2.5) does not have λ in its numerator. Hence, when the entry Ψ 2 * (λ) is excluded from the system, no apparent singularities appear. Therefore, in the case (2.5), the entry Ψ 1 * (λ) of the first row of the matrix function Ψ(λ) satisfies a linear 2nd order Fuchsian ODE with 4 singular points without any apparent singularity and therefore is equivalent to GHE. It is, in fact, straightforward to check that the function satisfies the general Heun equation in its canonical form (1.1) Observe that the expression for the accessory parameter ν in (2.10) besides the pole position and the local monodromies, involves the coefficient b 3 in the parameterization of the entry A 11 . Thus, taking into account formula for b 3 in (A.3), we see that the accessory parameter ν is determined by the free coefficient c 0 (or c −2 in the case (2.8 -see (A.4))) in the Laurent expansion of the sixth Painlevé transcendent.
For the coefficient matrix (2.6), corresponding to δ = 1, σ = −1 a similar statement is valid for the entries of the second row of Ψ(λ), In the case (2.7) corresponding to δ = 1, This completes the proof of Theorem 2.1.

Riemann-Hilbert problem approach to the Heun equation
Main result of this section is the formulation of the RH problem for the general Heun functions in the generic case We shall start, following closely references [5,17], with the standard definition of the monodromy data for Fuchsian system (2.1), (2.2) and with the related Riemann-Hilbert problem for the sixth Painlevé equation.

Monodromy data
Let λ 1 = 0 and λ 3 = 1. Then fix a point λ 2 = x ∈ C\{0, 1, ∞}, choose a base point λ 0 ∈ C\{0, 1, x, ∞} and cut the complex plane along the segments Encircle the points λ 1 = 0, λ 2 = x and λ 3 = 1 using non-intersecting circles C j , j = 1, 2, 3. Denote γ the graph and orient it as in Fig. 1. Denote also D j , j = 1, 2, 3 the interiors of the circles C j and D ∞ the domain The domains D j are assigned to the principal branches of the Frobenius (canonical) solutions to (2.1) at the Fuchsian singular points defined by the conditions Here, the branches of (λ − λ j ) α j , j = 1, 2, 3, and λ −δ are fixed by the condition Given a pair of the characteristic exponents (α j , −α j ), the matrix of eigenvectors T j ∈ SL(2, C) is determined up to a right diagonal factor. In contrast, the Frobenius solution Ψ ∞ (λ) is normalized and therefore, as soon as the pair (−δ, δ) is fixed, it is determined uniquely.
The matrices of the local monodromy are defined as the branch matrices of the Frobenius solutions Introduce also the connection matrices between Frobenius solutions at infinity and at the finite singularities Similar to Ψ j (λ), the connection matrices E j are determined modulo arbitrary right diagonal factors. In contrast, the monodromy matrices M j , are determined uniquely. The monodromy matrices are the branching matrices of the solution Ψ ∞ (λ) at the singular points λ j , j = 1, 2, 3; namely, one has that Together with the matrix they generate the monodromy group of equation (2.1) and are subject of one (cyclic) constraint Given the local monodromies, each of the monodromy matrices M j depends on 2 parameters. The total set of the 6 parameters determining the monodromy matrices M j , j = 1, 2, 3, ∞, is subject to a system of 3 scalar constraints. Thus the parameter set of the monodromy data involves generically 3 parameters. One of these parameters corresponds to the constant factor κ 0 determining the auxiliary function κ -see (A.1), (A.2). This is a reflection of the possible conjugation of A(λ) by a constant diagonal matrix -the action which does not affect the zero of A 12 (λ), i.e., the P VI function y(x). Neglecting this auxiliary parameter, the space of essential monodromy data, M, is invariant with respect to an overall conjugation by a diagonal matrix and can be identified with an algebraic variety -the monodromy surface, of dimension 2 (see below equation (3.6)). At the same time, the full space of monodromy data, can be represented as In [17], M. Jimbo has proposed a parameterization of the 2-dimensional monodromy surface M by the trace coordinates invariant with respect to the overall diagonal conjugation. Namely, letting one finds the relation between all these parameters for a 2-dimensional surface called the Fricke cubic According to [16], apart from the singular points of the surface (3.6), the monodromy matrices can be written explicitly in terms of the variables t ij . Exact formulae can be found in [17] and [16]. Each point of the surface M represents an isomonodromic family of equations (2.1) which in turns generates a solution y(x) of the Painlevé VI equation. Hence, the P VI transcendents can be parameterized by the points of M. In fact, at generic points of the Fricke cubic one can use any pair of the parameters t ij or σ ij to parameterize the set of the corresponding Painlevé functions. For instance, one can choose, so that we have the parameterization of the P VI functions by the pair (t, s), y ≡ y(x; t, s).
It also should be mentioned that some of the physically important solutions, e.g., the so-called classical solutions to P VI , correspond to non-generic points of the monodromy data set and for their parameterization one can use the full monodromy space M. We refer to [13] for more detail on this issue.

Riemann-Hilbert problem for P VI
The inverse monodromy problem, i.e., the problem of reconstruction of the function Ψ, and hence of the corresponding Painlevé function y(x), from their monodromy data is formulated as a Riemann-Hilbert (RH) problem. The direct and inverse monodromy problems associated with the equation P VI were studied by several authors. We mention here the pioneering paper [17], and subsequent papers [2,3,13].
We shall now formulate precisely the Riemann-Hilbert problem corresponding to the inverse monodromy problem for Fuchsian 2 × 2 system (2.1).
Proposition 3.4. Conversely, if for given x, δ, α, and matrices M j , E j the RH Problem 3.1 is solvable then the function Ψ(λ) satisfies the Fuchsian system (2.1) whose Frobenius solutions are determined by the solution Ψ(λ) of the RH problem according to the equations (3.9) (read backwards) and whose monodromy data coincide with the given RH data.

Riemann-Hilbert problem for the Heun function
Main result of this section states that the RH problem for the Heun function coincides with that for P VI supplemented by the additional condition of triangularity of the sub-leading term of the asymptotic expansion of Ψ(λ) as λ → ∞. Again, we consider the non-resonant case 2α 1 , 2α 2 , 2α 3 , 2δ / ∈ Z. On the other hand, the form of the coefficient matrix (2.5) with a 3 = −δ implies the following asymptotics of the solution to the linear ODE Ψ λ = AΨ, where the main difference from the asymptotic parameters in (3.10), (A.5) is the lower-triangular structure of the coefficient ψ 1 , We point out that the lower-triangular structure of ψ 1 implies the lower-triangular structure of the O λ −1 -term in the expansion (3.12). All other principal analytic properties of the limiting function Ψ(λ) including the leading order asymptotics at the singular points and the monodromy properties coincide with those of the function Ψ(λ) at the regular points of the Painlevé transcendent located in a sufficiently small neighborhood of the pole x = a.
Let us show that, conversely, this RH problem leads to the structure (2.5) of the coefficient matrix A(λ). Let Ψ(λ) be a unique solution of the RH Problem 3.5. First, det Ψ(λ) ≡ 1 since this determinant is piecewise holomorphic, continuous across the graph γ, bounded at λ j , j = 1, 2, 3, and at the nodes of the graph γ and approaches the unit as λ → ∞. Consider now the function A(λ) = Ψ λ Ψ −1 . It is piecewise holomorphic, continuous across γ, has simple poles at λ = λ j , j = 1, 2, 3, and λ = ∞, is bounded at the nodes of γ and therefore it is rational.
All these properties of A(λ) imply its structure given in (2.5). Thus, using (2.9), the first row of Ψ(λ) determines a fundamental system of solutions to GHE (2.10). We have proved the following Proposition 3.6. Solution of the RH Problem 3.5, if it exists, determines a fundamental system of solutions to GHE (2.10) with the prescribed monodromy properties.
The accessory parameter ν in (2.10) is also determined via the RH Problem 3.5. Indeed, ν can be extracted from the asymptotics of Ψ(λ) at infinity. Namely, the parameter d 1 , i.e., the diagonal part of the term O λ −1 of the asymptotic expansion of Ψλ δσ 3 , determines the coefficient b 3 in the coefficient matrix (2.5) and hence the free coefficient c 0 in the Laurent expansion (2.3) and the accessory parameter ν, Remark 3.7. The condition (2) of the RH Problem 3.5 can be in fact thought of as an addition to cyclic relation (3.5) restriction on the monodromy matrices {M j } which would also involve the point a. This restriction can be formulated in terms of the sixth Painlevé transcendent in two different but equivalent ways. Firstly, introducing on the monodromy surface M coordinates t and s (see (3.7)), let y(x) ≡ y(x; t, s) be the corresponding sixth Painlevé function (cf. (3.8)). Then the point a must be one of the (σ = +1) poles of y(x; t, s). Alternatively, assuming the position a of the pole of y(x) to be a free parameter, one can parameterize y(x) by the pair 1 (t, a), y(x) ≡ y(x; t, a). Then, the second monodromy data s becomes the function of (t, a) which can be described implicitly as follows. Note that together with s, the coefficient c 0 in the Laurent expansion (2.3) and the accessory parameter ν also become the functions of t and a, s ≡ s(t, a), c 0 ≡ c 0 (t, a), ν ≡ ν(t, a).
At the same time, the RH Problem 3.1 determines the solution Ψ(λ) and hence all the objects related to it, specifically the coefficients of its expansion (3.10) at λ = ∞, as the functions of the point on the monodromy surface, that is as the functions of t and s. In particular, we have that d 1 = d 1 (x; t, s). Using now the second equation in (3.13), the function s(t, a) can be defined implicitly via the equation (3.14) Remark 3.8. Excluding the parameter d 1 from the second and third equations in (3.13), we obtain the formula relating the accessory parameter ν(t, a) and the free coefficient c 0 (t, a) in the Laurent expansion (2.3), As it has already been mentioned before, this relation has been already found in [20] using a heuristic technique based on the remarkable connection (also discovered in [20]) between the classical conformal blocks and the sixth Painlevé equation.
Remark 3.9. In this section we considered the regular case, i.e., δ = 0, 1 2 and σ = +1 only. Our arguments, however, can be easily extended to the generic singular case, i.e., δ = 0, 1 2 , 1 and σ = −1, and to the special singular cases, i.e., δ = 1 and σ = −1 and δ = 1 2 . One only needs, before producing the relevant analogs of the RH problem 3.5, to make the preliminary Schlesinger transformations of the RH Problem 3.1 with the gauge matrices R(λ) discussed in details in Sections 2.3.2, 2.3.3 and 2.3.4. Remark 3.10. In this paper we are dealing with the poles of y(x). Similar results concerning the reduction of the RH Problem 3.1 to a Riemann-Hilbert problem for Heun equation can be obtained for two other critical values of y(x), i.e., when a is either a zero of y(x) or y(a) = 1.
We believe that the Riemann-Hilbert technique we are developing here can be used to study effectively the Heun functions. In particular, the relation (3.15) allows one to get nontrivial information about the accessory parameter ν(t, a). In particular, using the known connection formulae for the pole distributions of the P VI equation [12] (obtained with the help of the isomonodromy RH Problem 3.1; see also [13] for complete list of the asymptotic connection formulae and the history of the question), one can obtain the asymptotic expansion of the Laurent coefficient c 0 (t, a) for the either large values of a or for small values of a or for the values of a close to 1. This in turn would yield the explicit formulae for the asymptotic behavior of the accessory parameter ν(t, a) as a ∼ ∞, a ∼ 0 and a ∼ 1. This question should be addressed in details in the future work on this subject. In the rest of this paper, we will demonstrate the usefulness of the RH Problem 3.5 in the study of another issue related to the Heun equations which is the construction of its explicit solutions.
Thus the set of reducible monodromy data form a 2-dimensional linear space. Following [5], we first simplify the RH jump graph replacing γ by the broken line [λ 1 , Fig. 2.
On the plane, make a cut along the broken line [λ 1 , Although until we reach Proposition 3.14 it is not really important, we remind that λ 1 = 0, λ 2 = x, and λ 3 = 1.
Observe the following properties of f (λ): Let us represent the solution Ψ(λ) as the product The function Φ(λ) thus has the following properties: Riemann-Hilbert Problem 3.11.
Proof . First of all, by the conventional arguments, if a solution to this problem exists, it is unique.
Next, upper triangularity of all jump matrices, see condition (3), means that the first column of Φ(λ) is single-valued and continuous across the broken line (λ 1 , λ 2 ) ∪ (λ 2 , λ 3 ). Condition (2) means that the first column is bounded at λ = λ j , j = 1, 2, 3. Then condition (1) yields that the entries of the first column are some polynomials of degree n and n − 1, respectively.
The choice of c n−1 made above implies the normalization of Y (λ) at infinity that complies with the condition (1). Thus the explicitly constructed function Y (λ) solves the RH Problem 3.11. Since its solution is unique, proof of proposition is completed.
The weight function g(λ) in (3.18) can be understood as a generalization of the hypergeometric weight, thus we call the polynomials p n (λ) and p n−1 (λ) determined by the RH Problem 3.11 the generalized Jacobi polynomials. Now, we are going to construct the generalized Jacobi polynomials explicitly and relate them to the polynomial solutions of the Heun equation, i.e., to the Heun polynomials.
To this end, we look for solution to the RH Problem 3.11 in the form where R(λ) is a matrix-valued polynomial. Using the jump properties of Φ(λ), we find the jump properties of the scalar function φ(λ), One of the solutions to this scalar jump problem is given explicitly Observe the behavior of φ(λ) (3.20) at the singularities Below, we use the coefficients φ k of the expansion of φ(λ) (3.20) near infinity closely related to the moments of the weight function g(λ), The left factor R(λ) is a polynomial matrix of the Schlesinger transformation at infinity [19] and for n ≥ 0 it can be found explicitly. For instance, if n = 0 : R(λ) = R 0 (λ) = I, π 0 (λ) ≡ 1, In particular, relations (3.22) imply that the RH Problem 3.11 for n = 0 is always solvable, cf. [5], while for n = 1, this problem is solvable if φ 1 = 0.
To formulate the result for any fixed n ≥ 2, introduce the following explicit form of the polynomial matrix R(λ) = R n (λ), 23) and the column vectors of the coefficients of the polynomial entries of R n (λ), We also need the Hankel matrix H n = {φ i+j−1 } i,j=1,n of the moments (3.21) along with its determinant n , Finally define the sequence of coefficients f k for asymptotic expansion of f (λ) = j=1,2,3 Then we have the following Proposition 3.13. RH Problem 3.11 is solvable if and only if n = 0. Its solution has the form (3.19) where the coefficients of the polynomial entries of the matrix R(λ) (3.23) are given by Proof . Requiring that the product (3.19) has the canonical asymptotics we find the following expansions in the second column of the product (12) : : Thus expressions for q Evaluating the prescribed terms at all orders from λ −1 to λ −n , we find equations for the vector coefficients r n and p n . Combining them into the matrix form, it follows H n p n + (φ n+1 , φ n+2 , . . . , φ 2n ) T = 0, H n r n = (0, . . . , 0, 1) T . If n = 0, the moment matrix H n is invertible, and the coefficient vectors p n , r n are computed by (3.24). If n = 0 then for the equations (3.26) on the vectors p n and r n , there are two alternatives. The first one implies that R(λ) does not exist. The second alternative implies an infinite number of R(λ) and therefore contradicts the uniqueness of solution to the RH Problem 3.11.
This completes the proof.
In the next proposition, we find the classical sixth Painlevé functions determined by the explicitly solvable RH Problem 3.11. For instance, in the simplest cases, if n = 0 : Proposition 3.14. If n n+1 = 0 then the RH Problem 3.11 determines the classical solution to P VI corresponding to the Ψ function with the prescribed reducible monodromy data in terms of the moment functions φ k , k = 1, . . . , φ 2n+2 as follows (3.27) Proof . According to (3.11), in order to find the Painlevé function y we have to compute two first coefficients (ψ 1 ) + and (ψ 2 ) + along with the coefficient d 1 of the asymptotic expansion of Ψ(λ) at λ → ∞, see (3.10). Using (3.25) and (3.26) again, we find Proof . The coefficient (ψ 1 ) + is computed in (3.28), (3.29) The RH Problem 3.11 transforms to the Heun RH Problem 3.5 if (ψ 1 ) + = 0 -condition (2) of the RH Problem 3.5. In virtue of (3.29) this happens only if n+1 = 0.
The equation is an equation on the variable λ 2 = x. This equation determines those a = x for which the Heun equation with the given α j , δ satisfying (3.16) admits polynomial solutions given by Φ 11 (λ). Alternatively, one can keep x = a free, then (3.30) would be equation on the parameters s 1 and s 3 of the monodromy matrices (3.17). This in turn would yield a restriction on the accessory parameter of the Heun equation which would guarantee the existence of its polynomial solutions for given a. Remark 3.17. There is a vast literature devoted to the exact solutions of the Heun equationsee work [21] and references therein. We also want to mention the works [28] and [30]. In [28], the indicated at the beginning of this paper relation of the Heun equation and the Lame equation has been used to obtain the solvable Heun equations and their polynomial solutions from Lame polynomials corresponding to the finite-gap Lame potentials. In [30], the hypergeometric type integral representations for the solutions of the Heun equation are found in the case when one of the singularities {0, 1, t, ∞} is apparent. It would be important to understand these explicit solutions within the Riemann-Hilbert formalism which we are developing in this article. dy dx = y 2 − y + 2z x(x − 1) , dz dx = 1 x(x − 1)y(y − 1)(y − x) z 2 x − 2y − 2xy + 3y 2 + zy(y − 1)(y − x)(y + x − 1) − α 2 1 x(y − 1) 2 (y − x) 2 + α 2 3 (x − 1)y 2 (y − x) 2 − α 2 2 x(x − 1)y 2 (y − 1) 2 + δ(δ − 1)y 2 (y − 1) 2 (y − x) 2 . (A.2) Finally, eliminating z, one arrives at the second order ODE for y y xx = 1 2 which is the classical equation P VI .