Middle Convolution and Heun's Equation

Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformati...

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Datum:	2009
1. Verfasser:	Takemura, K.
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spelling	irk-123456789-1491662019-02-20T01:27:10Z Middle Convolution and Heun's Equation Takemura, K. Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation. 2009 Article Middle Convolution and Heun's Equation / K. Takemura // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 33 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34M35; 33E10; 34M55 http://dspace.nbuv.gov.ua/handle/123456789/149166 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.
format	Article
author	Takemura, K.
spellingShingle	Takemura, K. Middle Convolution and Heun's Equation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Takemura, K.
author_sort	Takemura, K.
title	Middle Convolution and Heun's Equation
title_short	Middle Convolution and Heun's Equation
title_full	Middle Convolution and Heun's Equation
title_fullStr	Middle Convolution and Heun's Equation
title_full_unstemmed	Middle Convolution and Heun's Equation
title_sort	middle convolution and heun's equation
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149166
citation_txt	Middle Convolution and Heun's Equation / K. Takemura // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 33 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT takemurak middleconvolutionandheunsequation
first_indexed	2025-07-12T21:33:04Z
last_indexed	2025-07-12T21:33:04Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 040, 22 pages Middle Convolution and Heun’s Equation? Kouichi TAKEMURA Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan E-mail: takemura@yokohama-cu.ac.jp Received November 26, 2008, in final form March 25, 2009; Published online April 03, 2009 doi:10.3842/SIGMA.2009.040 Abstract. Heun’s equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun’s equation. Key words: Heun’s equation; the space of initial conditions; the sixth Painlevé equation; middle convolution 2000 Mathematics Subject Classification: 34M35; 33E10; 34M55 1 Introduction Heun’s equation is a standard form of a second-order Fuchsian differential equation with four singularities, and it is given by d2y dz2 + ( γ z + δ z − 1 + ε z − t ) dy dz + αβz − q z(z − 1)(z − t) y = 0, (1.1) with the condition γ + δ + ε = α + β + 1. The parameter q is called an accessory parameter. Although the local monodromy (local expo- nent) is independent of q, the global monodromy (e.g. the monodromy on the cycle enclosing two singularities) depends on q. Some properties of Heun’s equation are written in the books [21, 23], but an important feature related with the theory of finite-gap potential for the case γ, δ, ε, α−β ∈ Z+ 1 2 (see [6, 24, 25, 26, 27, 28, 29, 31] etc.), which leads to an algorithm to calculate the global monodromy explicitly for all q, is not written in these books. The sixth Painlevé equation is a non-linear ordinary differential equation written as d2λ dt2 = 1 2 ( 1 λ + 1 λ− 1 + 1 λ− t )( dλ dt )2 − ( 1 t + 1 t− 1 + 1 λ− t ) dλ dt + λ(λ− 1)(λ− t) t2(t− 1)2 { (1− θ∞)2 2 − θ2 0 2 t λ2 + θ2 1 2 (t− 1) (λ− 1)2 + (1− θ2 t ) 2 t(t− 1) (λ− t)2 } . (1.2) A remarkable property of this differential equation is that the solutions do not have movable singularities other than poles. It is known that the sixth Painlevé equation is obtained by monodromy preserving deformation of Fuchsian system of differential equations, d dz ( y1 y2 ) = ( A0 z + A1 z − 1 + At z − t )( y1 y2 ) , A0, A1, At ∈ C2×2. ?This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The full collection is available at http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html mailto:takemura@yokohama-cu.ac.jp http://dx.doi.org/10.3842/SIGMA.2009.040 http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html 2 K. Takemura See Section 2 for expressions of the elements of the matrices A0, A1, At. By eliminating y2 we have second-order differential equation for y1, which have an additional apparent singularity z = λ other than {0, 1, t,∞} for generic cases, and the point λ corresponds to the variable of the sixth Painlevé equation. For details of monodromy preserving deformation, see [10]. In this paper we investigate the condition that the second-order differential equation for y1 is written as Heun’s equation. To get a preferable answer, we introduce the space of initial conditions for the sixth Painlevé equation which was discovered by Okamoto [18] to construct a suitable defining variety for the set of solutions to the (sixth) Painlevé equation. For Fuchsian systems of differential equations and local systems on a punctured Riemann sphere, Dettweiler and Reiter [2, 3] gave an algebraic analogue of Katz’ middle convolution functor [12]. Filipuk [5] applied them for the Fuchsian systems with four singularities, obtained an explicit relationship with the symmetry of the sixth Painlevé equation, and the author [30] calculated the corresponding integral transformation for the Fuchsian systems with four singu- larities. The middle convolution is labeled by a parameter ν, and we have two values which leads to non-trivial transformation on 2×2 Fuchsian system with four singularities (see Section 4). In this paper we consider the middle convolution which is a different value of the parameter ν from the one discussed in [5, 30]. We will also study the relationship between middle convolution and Heun’s equation. For special cases, the integral transformation raised by the middle convolution turns out to be a transformation on Heun’s equation, and we investigate these cases. Note that the description by the space of initial conditions for the sixth Painlevé equation is favorable. The integral transformation of Heun’s equation is applied for the study of novel solutions, which we will discuss in a separated publication. If the parameter of the middle convolution is a negative integer, then the integral transformation changes to a successive differential, and a transforma- tion defined by a differential operator on Heun’s equation was found in [29] as a generalized Darboux transformation (Crum–Darboux transformation). Hence the integral transformation on Heun’s equation can be regarded as a generalization of the generalized Darboux transformation, which is related with the conjectual duality by Khare and Sukhatme [15]. Special functions of the isomonodromy type including special solutions to the sixth Painlevé equation have been studied actively and they are related with various objects in mathematics and physics [16, 32]. On the other hand, special functions of Fuchsian type including special solutions to Heun’s equation are also interesting objects which are related with general relativity and so on. This paper is devoted to an attempt to clarify both sides of viewpoints. This paper is organized as follows: In Section 2, we fix notations for the Fuchsian system with four singularities. In Section 3, we define the space of initial conditions for the sixth Painlevé equation and observe that Heun’s equation is obtained from the Fuchsian equation by restricting to certain lines in the space of initial conditions. In Section 4, we review results on the middle convolution and construct integral transformations. In Section 5, we investigating relationship among the middle convolution, integral transformations of Heun’s equation and the space of initial conditions. In Section 6, we consider the case that the parameter on the middle convolution is integer. In the appendix, we describe topics which was put off in the text. 2 Fuchsian system of rank two with four singularities We consider a system of ordinary differential equations, dY dz = A(z)Y, A(z) = A0 z + A1 z − 1 + At z − t = ( a11(z) a12(z) a21(z) a22(z) ) , Y = ( y1 y2 ) , (2.1) where t 6= 0, 1, A0, A1, At are 2 × 2 matrix with constant elements. Then equation (2.1) is Fuchsian, i.e., any singularities on the Riemann sphere C ∪ {∞} are regular, and it may Middle Convolution and Heun’s Equation 3 have regular singularities at z = 0, 1, t,∞ on the Riemann sphere C ∪ {∞}. Exponents of equation (2.1) at z = 0 (resp. z = 1, z = t, z = ∞) are described by eigenvalues of the matrix A0 (resp. A1, At, −(A0 +A1 +At)). By the transformation Y → zn0(z− 1)n1(z− t)ntY , the system of differential equations (2.1) is replaced as A(z) → A(z) + (n0/z + n1/(z − 1) + n2/(z − t))I (I: unit matrix), and we can transform equation (2.1) to the one where one of the eigenvalues of Ai is zero for i ∈ {0, 1, t} by putting −ni to be one of the eigenvalues of the original Ai. If the exponents at z = ∞ are distinct, then we can normalize the matrix −(A0 + A1 + At) to be diagonal by a suitable gauge transformation Y → GY , A(z) → GA(z)G−1. In this paper we assume that one of the eigenvalues of Ai is zero for i = 0, 1, t and the matrix −(A0 + A1 + At) is diagonal, and we set A∞ = −(A0 + A1 + At) = ( κ1 0 0 κ2 ) . (2.2) By eliminating y2 in equation (2.1), we have a second-order linear differential equation, d2y1 dz2 + p1(z) dy1 dz + p2(z)y1 = 0, p1(z) = −a11(z)− a22(z)− d dza12(z) a12(z) , p2(z) = a11(z)a22(z)− a12(z)a21(z)− d dz a11(z) + a11(z) d dza12(z) a12(z) . (2.3) Set Ai = ( a (i) 11 a (i) 12 a (i) 21 a (i) 22 ) , (i = 0, 1, t). (2.4) It follows from equation (2.2) that a (0) 12 + a (1) 12 + a (t) 12 = 0, a (0) 21 + a (1) 21 + a (t) 21 = 0. Hence a12(z) and a21(z) are expressed as a12(z) = k1z + k2 z(z − 1)(z − t) , a21(z) = k̃1z + k̃2 z(z − 1)(z − t) , and we have a (0) 12 + a (1) 12 + a (t) 12 = 0, (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 = −k1, ta (0) 12 = k2, a (0) 21 + a (1) 21 + a (t) 21 = 0, (t + 1)a(0) 21 + ta (1) 21 + a (t) 21 = −k̃1, ta (0) 21 = k̃2. If k1 = k2 = 0, then y1 satisfies a first-order linear differential equation, and it is integrated easily. Hence we assume that (k1, k2) 6= (0, 0). Then it is shown that two of a (0) 12 , a (1) 12 , a (t) 12 , (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 cannot be zero. We set λ = −k2/k1 (k1 6= 0) and λ = ∞ (k1 = 0). The condition that none of a (0) 12 , a (1) 12 , a (t) 12 nor (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 is zero is equivalent to that λ 6= 0, 1, t,∞, and the condition a (0) 12 = 0 (resp. a (1) 12 = 0, a (t) 12 = 0, (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 = 0) is equivalent to λ = 0 (resp. λ = 1, λ = t, λ = ∞). We consider the case λ 6= 0, 1, t,∞, i.e., the case a (0) 12 6= 0, a (1) 12 6= 0, a (t) 12 6= 0, (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 6= 0. Let θ0 (resp. θ1, θt) and 0 be the eigenvalues of A0 (resp. A1, At). Then we can set A0, A1, At as A0 = ( u0 + θ0 −w0 u0(u0 + θ0)/w0 −u0 ) , A1 = ( u1 + θ1 −w1 u1(u1 + θ1)/w1 −u1 ) , At = ( ut + θt −wt ut(ut + θt)/wt −ut ) , (2.5) 4 K. Takemura by introducing variables u0, w0, u1, w1, ut, wt. By taking trace of equation (2.2), we have the relation θ0 +θ1 +θt +κ1 +κ2 = 0. We set θ∞ = κ1−κ2, then we have κ1 = (θ∞−θ0−θ1−θt)/2, κ2 = −(θ∞ + θ0 + θ1 + θt)/2. We determine u0, u1, ut, w0, w1, wt so as to satisfy equation (2.2) and the following relations: a12(z) = −w0 z − w1 z − 1 − wt z − t = k(z − λ) z(z − 1)(z − t) , a11(λ) = u0 + θ0 λ + u1 + θ1 λ− 1 + ut + θt λ− t = µ, (see [11]). Namely, we solve the following equations for u0, u1, ut, w0, w1, wt: −w0 − w1 − wt = 0, w0(t + 1) + w1t + wt = k, −w0t = −kλ, u0(u0 + θ0)/w0 + u1(u1 + θ1)/w1 + ut(ut + θt)/wt = 0, u0 + θ0 + u1 + θ1 + ut + θt = −κ1, −u0 − u1 − ut = −κ2, (u0 + θ0)/λ + (u1 + θ1)/(λ− 1) + (ut + θt)/(λ− t) = µ. (2.6) The linear equations for w0, w1, wt are solved as w0 = kλ t , w1 = −k(λ− 1) t− 1 , wt = k(λ− t) t(t− 1) . (2.7) By the equations which are linear in u0, u1 and ut, we can express u1 + θ1 and ut + θt as linear functions in u0. We substitute u1 + θ1 and ut + θt into a quadratic equation in u0, u1 and ut. Then the coefficient of u2 0 disappears, and u0, u1, ut are solved as u0 = −θ0 + λ tθ∞ [λ(λ− 1)(λ− t)µ2 + {2κ1(λ− 1)(λ− t)− θ1(λ− t) − tθt(λ− 1)}µ + κ1{κ1(λ− t− 1)− θ1 − tθt}], u1 = −θ1 − λ− 1 (t− 1)θ∞ [λ(λ− 1)(λ− t)µ2 + {2κ1(λ− 1)(λ− t) + (θ∞ − θ1)(λ− t) − tθt(λ− 1)}µ + κ1{κ1(λ− t + 1) + θ0 − (t− 1)θt}], ut = −θt + λ− t t(t− 1)θ∞ [λ(λ− 1)(λ− t)µ2 + {2κ1(λ− 1)(λ− t)− θ1(λ− t) + t(θ∞ − θt)(λ− 1)}µ + κ1{κ1(λ− t + 1) + θ0 + (t− 1)(θ∞ − θt)}]. (2.8) We denote the Fuchsian system of differential equations dY dz = ( A0 z + A1 z − 1 + At z − t ) Y, Y = ( y1 y2 ) , (2.9) with equations (2.5), (2.7), (2.8) by DY (θ0, θ1, θt, θ∞;λ, µ; k). Then the second-order differential equation (2.3) is written as d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 1− θt z − t − 1 z − λ ) dy1 dz + ( κ1(κ2 + 1) z(z − 1) + λ(λ− 1)µ z(z − 1)(z − λ) − t(t− 1)H z(z − 1)(z − t) ) y1 = 0, H = 1 t(t− 1) [λ(λ− 1)(λ− t)µ2 − {θ0(λ− 1)(λ− t) + θ1λ(λ− t) + (θt − 1)λ(λ− 1)}µ + κ1(κ2 + 1)(λ− t)], (2.10) Middle Convolution and Heun’s Equation 5 which we denote by Dy1(θ0, θ1, θt, θ∞;λ, µ). This equation has regular singularities at z = 0, 1, t, λ,∞. Exponents of the singularity z = λ are 0, 2, and it is apparent (non-logarithmic) singularity. Note that the differential equations dλ dt = ∂H ∂µ , dµ dt = −∂H ∂λ (2.11) describe the condition for monodromy preserving deformation of equation (2.3) with respect to the variable t. By eliminating the variable µ in equation (2.11), we have the sixth Painlevé equation on the variable λ (see equation (1.2)). See [20] on equations (2.3), (2.10) and (2.11). We consider realization of the Fuchsian system (equation (2.1)) for the case λ = 0, 1, t,∞ in the appendix. 3 The space of initial conditions for the sixth Painlevé equation and Heun’s equation In this section, we introduce the space of initial conditions for the sixth Painlevé equation, restrict the variables of the space of initial conditions E(t) to certain lines, and we obtain Heun’s equation. The space of initial conditions was introduced by Okamoto [18], which is a suitable defining variety for the set of solutions to the Painlevé system. In [22], Shioda and Takano studied the space of initial conditions further for the sixth Painlevé system (equation (2.11)) to study roles of holomorphy on the Hamiltonian. It was also constructed as a moduli space of parabolic connections by Inaba, Iwasaki and Saito [8, 9]. Here we adopt the coordinate of initial coordinate by Shioda and Takano [22] (see also [33]). The space of initial condition E(t) is defined by patching six copies U0 = {(q0, p0)}, U1 = {(q1, p1)}, U2 = {(q2, p2)}, U3 = {(q3, p3)}, U4 = {(q4, p4)}, U∞ = {(q∞, p∞)}, (3.1) of C2 for fixed (t; θ0, θ1, θt, θ∞), and the rule of patching is defined by q0q∞ = 1, q0p0 + q∞p∞ = −κ1, (U0 ∩ U∞), q0p0 + q1p1 = θ0, p0p1 = 1, (U0 ∩ U1), (q0 − 1)p0 + q2p2 = θ1, p0p2 = 1, (U0 ∩ U2), (q0 − t)p0 + q3p3 = θt, p0p3 = 1, (U0 ∩ U3), q∞p∞ + q4p4 = 1− θ∞, p∞p4 = 1, (U∞ ∩ U4). (3.2) The variables (λ, µ) of the sixth Painlevé system (see equation (2.11)) are realized as q0 = λ, p0 = µ in U0. We define complex lines in the space of initial conditions as follows: L0 = {(0, p0)} ⊂ U0, L1 = {(1, p0)} ⊂ U0, Lt = {(t, p0)} ⊂ U0, L∞ = {(0, p∞)} ⊂ U∞, L∗0 = {(q1, 0)} ⊂ U1, L∗1 = {(q2, 0)} ⊂ U2, L∗t = {(q3, 0)} ⊂ U3, L∗∞ = {(q4, 0)} ⊂ U4. (3.3) Set U q0 6=0,1,t 0 = U0 \ (L0 ∪ L1 ∪ Lt). 6 K. Takemura Then the space of initial conditions E(t) is a direct sum of the sets U q0 6=0,1,t 0 , L0, L1, Lt, L∞, L∗0, L∗1, L∗t , L∗∞. If (λ, µ) ∈ U q0 6=0,1,t 0 , then λ 6= 0, 1, t,∞ and equation (2.10) has five regular singularities {0, 1, t, λ,∞}. Although equation (2.6) was considered on the set U q0 6=0,1,t 0 , we may consider realization of a second-order differential equation as equation (2.10) on the space of initial conditions E(t). On the lines L0, L1, Lt, equation (2.10) is realized by setting λ = 0, 1, t, and the equation is written in the form of Heun’s equation d2y1 dz2 + ( −θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy1 dz + κ1(κ2 + 1)z + tθ0µ z(z − 1)(z − t) y1 = 0, (3.4) d2y1 dz2 + ( 1− θ0 z + −θ1 z − 1 + 1− θt z − t ) dy1 dz + κ1(κ2 + 1)(z − 1) + (1− t)θ1µ z(z − 1)(z − t) y1 = 0, (3.5) d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + −θt z − t ) dy1 dz + κ1(κ2 + 1)(z − t) + t(t− 1)θtµ z(z − 1)(z − t) y1 = 0, (3.6) respectively. Note that if θ0θ1θt 6= 0 then we can realize all values of accessory parameter as varying µ. For the case θ0θ1θt = 0, we should consider other realizations. To realize equation (2.10) on the line L∗0, we change the variables (λ, µ) into the ones (q1, p1) on equation (2.10) by applying relations λµ + q1p1 = θ0, µp1 = 1. Then we have d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 1− θt z − t − 1 z + p1(p1q1 − θ0) ) dy1 dz + κ1(κ2 + 1)z2 + (tq1 − θ0(θt + tθ1 + p1 pol1)z + (p1q1 − θ0)(−t− p1 pol2) z(z − 1)(z − t)(z + p1(p1q1 − θ0)) y1 = 0, where pol1 and pol2 are polynomials in p1, q1, t, θ0, θ1, θt, θ∞. By setting p1 = 0, we obtain d2y1 dz2 + ( −θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy1 dz + κ1(κ2 + 1)z2 + (tq1 − θ0(θt + tθ1))z + tθ0 z2(z − 1)(z − t) y1 = 0. (3.7) Since the exponents of equation (3.7) at z = 0 are 1 and θ0, we consider gauge-transformation v1 = z−1y1 to obtain Heun’s equation, and we have d2v1 dz2 + ( 2− θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dv1 dz + (κ1 + 1)(κ2 + 2)z − q z(z − 1)(z − t) v1 = 0, q = −tq1 + (θ0 − 1){t(θ1 − 1) + θt − 1}. (3.8) To realize the second-order Fuchsian equation on the line L∗1, we change the variables (λ, µ) into the ones (q2, p2), substitute p2 = 0 into equation (2.10) and set v1 = (z − 1)−1y1. Then v1 satisfies the following equation; d2v1 dz2 + ( 1− θ0 z + 2− θ1 z − 1 + 1− θt z − t ) dv1 dz + (κ1 + 1)(κ2 + 2)(z − 1)− q z(z − 1)(z − t) v1 = 0, q = (t− 1)q2 − (θ1 − 1){(1− t)(θ0 − 1) + θt − 1}. (3.9) The second-order Fuchsian equation on the line L∗t is realized as d2v1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 2− θt z − t ) dv1 dz + (κ1 + 1)(κ2 + 2)(z − t)− q z(z − 1)(z − t) v1 = 0, Middle Convolution and Heun’s Equation 7 q = t(1− t)q3 − (θt − 1)((t− 1)(θ0 − 1) + t(θ1 − 1)), (3.10) by setting p3 = 0 and v1 = (z − t)−1y1. We investigate equation (2.10) on the line L∞. We change the variables (λ, µ) into the ones (q∞, p∞) on equation (2.10) by applying relations λq∞ = 1, λµ + q∞p∞ = −κ1, and substitute q∞ = 0. Then we have d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy1 dz + κ1(κ2 + 2)z − q z(z − 1)(z − t) y1 = 0, q = (θ∞ − 1)p∞ + κ1(t(κ2 + θt + 1) + κ2 + θ1 + 1). Note that the exponents at z = ∞ are κ1 and κ2 + 2. To realize equation (2.10) on the line L∗∞, we change the variables (λ, µ) into the ones (q4, p4) on equation (2.10) by applying relations λq∞ = 1, λµ + q∞p∞ = −κ1, q∞p∞ + q4p4 = 1 − θ∞, p∞p4 = 1, substitute p4 = 0. We obtain d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy1 dz + (κ1 + 1)(κ2 + 1)z − q z(z − 1)(z − t) y1 = 0, q = −q4 + (κ2 + 1)(t(κ1 + θt) + κ1 + θ1). (3.11) The exponents at z = ∞ are κ1 +1 and κ2 +1, which are different from the case of the line L∞. The Fuchsian system DY (θ0, θ1, θt, θ∞;λ, µ; k) is originally defined on the set U q0 6=0,1,t 0 . We try to consider realization of Fuchsian system (equation (2.1)) on the lines L0, L∗0, L1, L∗1, Lt, L∗t , L∞, L∗∞ in the appendix. 4 Middle convolution First, we review an algebraic analogue of Katz’ middle convolution functor developed by Det- tweiler and Reiter [2, 3], which we restrict to the present setting. Let A0, A1, At be matrices in C2×2. For ν ∈ C, we define the convolution matrices B0, B1, Bt ∈ C6×6 as follows: B0 =  A0 + ν A1 At 0 0 0 0 0 0  , B1 =  0 0 0 A0 A1 + ν At 0 0 0  , Bt =  0 0 0 0 0 0 A0 A1 At + ν  . (4.1) Let z ∈ C\{0, 1, t}, γp (p ∈ C) be a cycle in C\{0, 1, t, z} turning the point w = p anti-clockwise whose fixed base point is o ∈ C \ {0, 1, t, z}, and [γp, γp′ ] = γpγp′γ −1 p γ−1 p′ be the Pochhammer contour. Proposition 1 ([3]). Assume that Y = t(y1(z), y2(z)) is a solution to the system of differential equations dY dz = ( A0 z + A1 z − 1 + At z − t ) Y. 8 K. Takemura For p ∈ {0, 1, t,∞}, the function U =  ∫ [γz ,γp] w−1y1(w)(z − w)νdw∫ [γz ,γp] w−1y2(w)(z − w)νdw∫ [γz ,γp] (w − 1)−1y1(w)(z − w)νdw∫ [γz ,γp] (w − 1)−1y2(w)(z − w)νdw∫ [γz ,γp] (w − t)−1y1(w)(z − w)νdw∫ [γz ,γp] (w − t)−1y2(w)(z − w)νdw  , satisfies the system of differential equations dU dz = ( B0 z + B1 z − 1 + Bt z − t ) U. (4.2) We set L0 =  Ker(A0) 0 0  , L1 =  0 Ker(A1) 0  , Lt =  0 0 Ker(At)  , L = L0 ⊕ L1 ⊕ Lt, K = Ker(B0) ∩Ker(B1) ∩Ker(Bt), (4.3) where L0, L1, Lt, K ⊂ C6 and 0 in equation (4.3) means the zero vector in C2. We fix an isomorphism between C6/(K + L) and Cm for some m. A tuple of matrices mcν(A) = (B̃0, B̃1, B̃t), where B̃p (p = 0, 1, t) is induced by the action of Bp on Cm ' C6/(K + L), is called an additive version of the middle convolution of (A0, A1, At) with the parameter ν. Let A0, A1, At be the matrices defined by equation (2.5). Then it is shown that if ν = 0, κ1, κ2 (resp. ν 6= 0, κ1, κ2) then dim C6/(K + L) = 2 (resp. dim C6/(K + L) = 3). If ν = 0, then the middle convolution is identity (see [3]). Hence the middle convolutions for two cases ν = κ1, κ2 may lead to non-trivial transformations on the 2 × 2 Fuchsian system with four singularities {0, 1, t,∞}. Filipuk [5] obtained that the middle convolution for the case ν = κ1 induce an Okamoto’s transformation of the sixth Painlevé system. We now calculate explicitly the Fuchsian system of differential equations determined by the middle convolution for the case ν = κ2. Note that the following calculation is analogous to the one in [30] for the case ν = κ1. If ν = κ2, then the spaces L0, L1, Lt, K are written as L0 = C  w0 u0 + θ0 0 0 0 0  , L1 = C  0 0 w1 u1 + θ1 0 0  , Lt = C  0 0 0 0 wt ut + θt  , K = C  0 1 0 1 0 1  . Middle Convolution and Heun’s Equation 9 Set S =  0 0 0 w0 0 0 0 0 1 u0 + θ0 0 0 0 0 0 0 w1 0 s41 s42 1 0 u1 + θ1 0 0 0 0 0 0 wt s61 s62 1 0 0 ut + θt  , s41 = µ(λ− t) + κ1 kκ1 , s61 = t(µ(λ− 1) + κ1) kκ1 , s42 = λ̃− λ λ(λ− 1)κ2 , s62 = t(λ̃− λ) λ(λ− t)κ2 , λ̃ = λ− κ2 µ− θ0 λ − θ1 (λ−1) − θt (λ−t) , (4.4) and Ũ = S−1U , where U is a solution to equation (4.2). Then det U = k2(λ̃ − λ)/(t(1 − t)κ2) and Ũ satisfies dŨ dz =  b11(z) b12(z) 0 0 0 0 b21(z) b22(z) 0 0 0 0 −(u0+θ0)θ∞t kκ1λz λ̃ λz 0 0 0 0 t kλz 0 0 κ2 z 0 0 1−t k(λ−1)(z−1) 0 0 0 κ2 z−1 0 t(1−t) k(λ−t)(z−t) 0 0 0 0 κ2 z−t  Ũ , where b11(z), . . . , b22(z) are calculated such that the system of differential equation dỸ dz = ( b11(z) b12(z) b21(z) b22(z) ) Ỹ , Ỹ = ( ũ1(z) ũ2(z) ) , coincides with the Fuchsian system DY (θ̃0, θ̃1, θ̃t, θ̃∞; λ̃, µ̃; k̃) (see equation (2.9)), where θ̃0 = θ0 − θ1 − θt − θ∞ 2 , θ̃1 = −θ0 + θ1 − θt − θ∞ 2 , θ̃t = −θ0 − θ1 + θt − θ∞ 2 , θ̃∞ = −θ0 − θ1 − θt + θ∞ 2 , λ̃ = λ− κ2 µ− θ0 λ − θ1 λ−1 − θt λ−t , µ̃ = κ2 + θ0 λ̃ + κ2 + θ1 λ̃− 1 + κ2 + θt λ̃− t + κ2 λ− λ̃ , k̃ = k. (4.5) The functions ũ1(z) and ũ2(z) are expressed as ũ1(z) = (u0 + θ0)u1(z)− kλ t u2(z) + (u1 + θ1)u3(z) + k(λ− 1) t− 1 u4(z) + (ut + θt)u5(z) + k(λ− t) t(1− t) u6(z), ũ2(z) = κ2λ(λ− 1)(λ− t) κ1(λ− λ̃) ( (λµ + κ1)(u0 + θ0) kλ u1(z)− λµ + κ1 t u2(z) + ((λ− 1)µ + κ1)(u1 + θ1) k(λ− 1) u3(z) + (λ− 1)µ + κ1 t− 1 u4(z) + ((λ− t)µ + κ1)(ut + θt) k(λ− t) u5(z) + (λ− t)µ + κ1 t(1− t) u6(z) ) . (4.6) Combining Proposition 1 with equation (4.6) and setting ỹ1(z) = ũ1(z), ỹ2(z) = ũ2(z), we have the following theorem by means of a straightforward calculation: 10 K. Takemura Theorem 1. Set κ1 = (θ∞ − θ0 − θ1 − θt)/2 and κ2 = −(θ∞ + θ0 + θ1 + θt)/2. If y1(z) is a solution to the Fuchsian equation Dy1(θ0, θ1, θt, θ∞;λ, µ), then the function Ỹ = t(ỹ1(z), ỹ2(z)) defined by ỹ1(z) = ∫ [γz ,γp] dy1(w) dw (z − w)κ2dw, (4.7) ỹ2(z) = κ2λ(λ− 1)(λ− t) k(λ− λ̃) ∫ [γz ,γp] {( dy1(w) dw − µy1(w) ) 1 λ− w + µ κ1 dy1(w) dw } (z − w)κ2dw, satisfies the Fuchsian system DY (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃; k) for p ∈ {0, 1, t,∞}, where λ̃ = λ− κ2 µ− θ0 λ − θ1 λ−1 − θt λ−t , µ̃ = κ2 + θ0 λ̃ + κ2 + θ1 λ̃− 1 + κ2 + θt λ̃− t + κ2 λ− λ̃ . (4.8) Since 0 = ∫ [γz ,γp] d dw (y1(w)(z − w)κ2) dw = ∫ [γz ,γp] dy1(w) dw (z − w)κ2dw + κ2 ∫ [γz ,γp] y1(w)(z − w)κ2−1dw, (4.9) we have Proposition 2 ([17]). If y1(z) is a solution to Dy1(θ0, θ1, θt, θ∞;λ, µ), then the function ỹ(z) = ∫ [γz ,γp] y1(w)(z − w)κ2−1dw, (4.10) satisfies Dy1(κ2 +θ0, κ2 +θ1, κ2 +θt, κ2 +θ∞; λ̃, µ̃) for p ∈ {0, 1, t,∞}, where λ̃ and µ̃ are defined in equation (4.8). Note that this proposition was obtained by Novikov [17] by another method. Kazakov and Slavyanov [14] essentially obtained this proposition by investigating Euler transformation of 2 × 2 Fuchsian systems with singularities {0, 1, t,∞} which are realized differently from DY (θ0, θ1, θt, θ∞;λ, µ, k). Let us recall the symmetry of the sixth Painlevé equation. It was essentially established by Okamoto [19] that the sixth Painlevé equation has symmetry of the affine Weyl group W (D(1) 4 ). More precisely, the sixth Painlevé system is invariant under the following transformations, which are involutive and satisfy Coxeter relations attached to the Dynkin diagram of type D (1) 4 , i.e. (si)2 = 1 (i = 0, 1, 2, 3, 4), sjsk = sksj (j, k ∈ {0, 1, 3, 4}), sjs2sj = s2sjs2 (j = 0, 1, 3, 4): θt θ∞ θ1 θ0 λ µ t s0 −θt θ∞ θ1 θ0 λ µ− θt λ−t t s1 θt 2− θ∞ θ1 θ0 λ µ t s2 κ1 + θt −κ2 κ1 + θ1 κ1 + θ0 λ + κ1 µ µ t s3 θt θ∞ −θ1 θ0 λ µ− θ1 λ−1 t s4 θt θ∞ θ1 −θ0 λ µ− θ0 λ t i i i i i 0 3 1 4 2 The map (θ0, θ1, θt, θ∞;λ, µ) 7→ (θ̃0, θ̃1, θ̃t, θ̃∞; λ̃, µ̃) determined by equation (4.5) coincides with the composition map s0s3s4s2s0s3s4, because (θ0, θ1, θt, θ∞;λ, µ) s0s3s47→ ( − θ0,−θ1,−θt, θ∞;λ, µ− θ0 λ − θ1 λ−1 − θt λ−t ) Middle Convolution and Heun’s Equation 11 s27→ ( − κ2 − θ0,−κ2 − θ1,−κ2 − θt, κ1; λ̃, µ− θ0 λ − θ1 λ−1 − θt λ−t ) s0s3s47→ ( κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃ ) . Therefore, if we know a solution to the Fuchsian system DY (θ0, θ1, θt, θ∞;λ, µ; k), then we have integral representations of solutions to the Fuchsian system DY (θ̃0, θ̃1, θ̃t, θ̃∞; λ̃, µ̃; k) obtained by the transformation s0s3s4s2s4s3s0. Note that the transformations si (i = 0, 1, 2, 3, 4) are extended to isomorphisms of the space of initial conditions E(t). We recall the middle convolution for the case ν = κ1. Proposition 3 ([30, Proposition 3.2]). If Y = t(y1(z), y2(z)) is a solution to the Fuch- sian system DY (θ0, θ1, θt, θ∞;λ, µ; k) (see equation (2.9)), then the function Ỹ = t(ỹ1(z), ỹ2(z)) defined by ỹ1(z) = ∫ [γz ,γp] { κ1y1(w) + (w − λ̃) dy1(w) dw } (z − w)κ1 w − λ dw, ỹ2(z) = −θ∞ κ2 ∫ [γz ,γp] dy2(w) dw (z − w)κ1dw, (4.11) satisfies the Fuchsian system DY (κ1 +θ0, κ1 +θ1, κ1 +θt,−κ2;λ+κ1/µ, µ; k) for p ∈ {0, 1, t,∞}. The parameters (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ) are obtained from the parameters (θ0, θ1, θt, θ∞;λ, µ) by applying the transformation s2. Note that the relationship the transfor- mation s2 was obtained by Filipuk [5] explicitly (see also [7]), and Boalch [1] and Dettweiler and Reiter [4] also obtained results on the symmetry of the sixth Painlevé equation and the middle convolution. 5 Middle convolution, integral transformations of Heun’s equation and the space of initial conditions In this section, we investigating relationship among the middle convolution, integral transfor- mations of Heun’s equation and the space of initial conditions. Kazakov and Slavyanov established an integral transformation on solutions to Heun’s equa- tion in [13], which we express in a slightly different form. Theorem 2 ([13]). Set (η − α)(η − β) = 0, γ′ = γ + 1− η, δ′ = δ + 1− η, ε′ = ε + 1− η, {α′, β′} = {2− η,−2η + α + β + 1}, q′ = q + (1− η)(ε + δt + (γ − η)(t + 1)). (5.1) Let v(w) be a solution to d2v dw2 + ( γ′ w + δ′ w − 1 + ε′ w − t ) dv dw + α′β′w − q′ w(w − 1)(w − t) v = 0. (5.2) Then the function y(z) = ∫ [γz ,γp] v(w)(z − w)−ηdw is a solution to d2y dz2 + ( γ z + δ z − 1 + ε z − t ) dy dz + αβz − q z(z − 1)(z − t) y = 0, (5.3) for p ∈ {0, 1, t,∞}. 12 K. Takemura Here we derive Theorem 2 by considering the limit λ → 0 in Proposition 2. Let us recall notations in Proposition 2. We consider the limit λ → 0 while fixing µ for the case θ0 6= 0 and θ0 + κ2 6= 0. Then we have λ̃ → 0 and µ̃ → (tθ0µ + κ2(t(κ1 + θt) + κ1 + θ1))/(t(κ2 + θ0)). Hence it follows from Proposition 2 and equation (3.4) that, if y(z) satisfies d2y(z) dz2 + ( −θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy(z) dz + κ1(κ2 + 1)z + tθ0µ z(z − 1)(z − t) y(z) = 0, (5.4) then the function ỹ(z) = ∫ [γz ,γp] y(w)(z − w)κ2−1dw, (5.5) satisfies d2ỹ(z) dz2 + ( −κ2 − θ0 z + 1− κ2 − θ1 z − 1 + 1− κ2 − θt z − t ) dỹ(z) dz + θ∞(1− κ2)z + t(κ2 + θ0) tθ0µ+κ2(t(κ1+θt)+κ1+θ1)) t(κ2+θ0) z(z − 1)(z − t)  ỹ(z) = 0. (5.6) By setting γ = −κ2 − θ0, δ = 1 − κ2 − θ1, ε = 1 − κ2 − θt, α = η = 1 − κ2, β = θ∞, q = −{tθ0µ+κ2(t(κ1+θt)+κ1+θ1))} and comparing with the standard form of Heun’s equation (equation (1.1)), we recover Theorem 2. Note that we can obtain the formula corresponding to the case θ0 = 0 (resp. θ0 + κ2 = 0) by considering the limit θ0 → 0 (resp. θ0 + κ2 → 0). The limit λ → 0 while fixing µ implies the restriction of the coordinate (λ, µ) to the line L0 in the space of initial conditions E(t), and the line L0 with the parameter (θ0, θ1, θt, θ∞;λ, µ) is mapped to the line L0 in the space of initial conditions with the parameter (κ2+θ0, κ2+θ1, κ2+θt, κ2 + θ∞; λ̃, µ̃) where λ̃ and µ̃ are defined in equation (4.8), because λ̃ → 0 and µ̃ converges by the limit. It follows from equations (5.4), (5.5), (5.6) that the integral transformation in Proposition 2 reproduces the integral transformation on Heun’s equations in Theorem 2 by restricting to the line L0. We can also establish that the line L1 (resp. Lt) in the space of initial conditions with the parameter (θ0, θ1, θt, θ∞;λ, µ) is mapped to the line L1 (resp. Lt) with the parameter (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) by taking the limit λ → 1 (resp. λ → t), and the integral transformation in Proposition 2 reproduces the integral transformation on Heun’s equations in Theorem 2. We discuss the restriction of the map (θ0, θ1, θt, θ∞;λ, µ) 7→ (κ2+θ0, κ2+θ1, κ2+θt, κ2+θ∞; λ̃, µ̃) to the line L∗∞. Let (q4, p4) (resp. (q̃4, p̃4)) be the coordinate of U4 for the parameters (θ0, θ1, θt, θ∞;λ, µ) (resp. (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) (see equations (3.1), (3.2)). Then we can express q̃4 and p̃4 by the variables q4 and p4. By setting p4 = 0, we have p̃4 = 0 and q̃4 = q4 − κ2(t(κ1 + θt − 1) + κ1 + θ1 − 1). Hence the line L∗∞ with the parameter (θ0, θ1, θt, θ∞;λ, µ) is mapped to the line L∗∞ with the parameter (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃). It follows from Proposition 2 that if y1(z) satisfies equation (3.11) then the function ỹ(z) defined by Proposition 2 satisfies Heun’s equation with the parameters γ = 1 − θ0 − κ2, δ = 1 − θ1 − κ2, ε = 1 − θt − κ2, α = 1 − κ2, β = 1 + θ∞, q = −q4−(1+t)κ2+(t(κ1+θt)+κ1+θ1)), and the integral representation reproduces Theorem 2 by setting η = α = 1− κ2. Therefore we have the following theorem: Theorem 3. Let X = L0, L1, Lt or L∗∞. By the map s0s3s4s2s4s3s0 : (θ0, θ1, θt, θ∞;λ, µ) 7→ (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) where λ̃ and µ̃ are defined in equation (4.8), the line X in the space of initial conditions with the parameter (θ0, θ1, θt, θ∞;λ, µ) is mapped to the line X in the space of initial conditions with the parameter (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) where λ̃ and µ̃ are defined in equation (4.8), and the integral transformation in Proposition 2 determined by the middle convolution reproduces the integral transformation on Heun’s equations in Theorem 2 by the restriction to the line X. Middle Convolution and Heun’s Equation 13 Note that if X = L∗0, L∗1, L∗t or L∞ then the image of the line X by the map s0s3s4s2s4s3s0 may not included in X with the parameter (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃). We consider restrictions of the middle convolution for the case ν = κ1 (see Proposition 3) to lines in the space of initial conditions. We discuss the restriction of the map (θ0, θ1, θt, θ∞;λ, µ) 7→ (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ) to the line L∗0. Let (q1, p1) (resp. (q̃1, p̃1)) be the coordinate of U1 for the parameters (θ0, θ1, θt, θ∞;λ, µ) (resp. (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ)). Then we can express q̃1 and p̃1 by the variables q1 and p1, and by setting p1 = 0 we have p̃1 = 0 and q̃1 = q1. Let y1(z) be a solution to the Fuchsian equation Dy1(θ0, θ1, θt, θ∞;λ, µ) for the case p1 = 0 and set v1(z) = z−1y1(z). Then v1(z) satisfies Heun’s equation written as equation (3.8). On the case p1 = 0, the integral representation (equation (4.11)) is written as ỹ1(z) = ∫ [γz ,γp] { κ1y1(w) + w dy1(w) dw } (z − w)κ1 w dw, (p ∈ {0, 1, t,∞}). We set ṽ1(z) = z−1ỹ1(z). By integration by parts we have ṽ1(z) = 1 z ∫ [γz ,γp] { κ1v1(w)(z − w)κ1 + d(wv1(w)) dw (z − w)κ1 } dw = 1 z ∫ [γz ,γp] { κ1(z − w)v1(w)(z − w)κ1−1 + κ1wv1(w)(z − w)κ1−1 } dw = κ1 ∫ [γz ,γp] v1(w)(z − w)κ1−1dw. (5.7) On the other hand, it follows from Proposition 3 and equation (3.8) that ṽ1(z) satisfies Heun’s equation with the parameters γ = 2 − θ0 − κ1, δ = 1 − θ1 − κ1, ε = 1 − θt − κ1, α = 1 − κ1, β = 2−θ∞, q = −tq1+(κ1+θ0−1){t(κ1+θ1−1)+κ1+θt−1}. Hence equation (5.7) reproduces Theorem 2 by setting η = α = 1 − κ1, We can also obtain similar results for L∗1, L∗t and L∗∞. Therefore we have the following theorem: Theorem 4. Let X = L∗0, L∗1, L∗t or L∗∞. By the map s2 : (θ0, θ1, θt, θ∞;λ, µ) 7→ (κ1 + θ0, κ1 +θ1, κ1 +θt,−κ2;λ+κ1/µ, µ), the line X in the space of initial conditions with the parameter (θ0, θ1, θt, θ∞;λ, µ) is mapped to the line X in the space of initial conditions with the parameter (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ), and the integral transformation in Proposition 3 determined by the middle convolution reproduces the integral transformation on Heun’s equations in Theorem 2 by the restriction to the line X. Note that if X = L0, L1, Lt or L∞ then the image of the line X by the map s2 may not included in X with the parameter (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ). 6 Middle convolution for the case that the parameter is integer On the case κ2 ∈ Z, the function in equation (4.10) containing the Pochhammer contour may be vanished, and we propose other expressions of solutions to Fuchsian equation Dy1(κ2+θ0, κ2+θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) in use of solutions to Dy1(θ0, θ1, θt, θ∞;λ, µ). We have the following proposition for the case κ2 ∈ Z<0, Proposition 4. (i) Let A0, A1, At be matrices in C2×2, and let B (ν) 0 , B (ν) 1 , B (ν) t ∈ C6×6 be the matrices defined in equation (4.1) for ν ∈ C. Assume that ν ∈ Z<0 and Y = t(y1(z), y2(z)) is a solution to the system of differential equations dY dz = ( A0 z + A1 z − 1 + At z − t ) Y. 14 K. Takemura Write ν = −1− n (n ∈ Z≥0). Then the function U =  (d/dz)n(z−1y1(z)) (d/dz)n(z−1y2(z)) (d/dz)n((z − 1)−1y1(z)) (d/dz)n((z − 1)−1y2(z)) (d/dz)n((z − t)−1y1(z)) (d/dz)n((z − t)−1y2(z))  , satisfies the system of differential equations dU dz = ( B (−1−n) 0 z + B (−1−n) 1 z − 1 + B (−1−n) t z − t ) U. (6.1) (ii) If κ2 ∈ Z<0 and Y = t(y1(z), y2(z)) is a solution to the Fuchsian system DY (θ0, θ1, θt, θ∞; λ, µ; k), then the function Ỹ = t(ỹ1(z), ỹ2(z)) defined by ỹ1(z) = ( d dz )−κ2 y1(z), ỹ2(z) = κ2λ(λ− 1)(λ− t) k(λ− λ̃) [{( d dz )−κ2 y1(z)− µ ( d dz )−κ2−1 y1(z) } 1 λ− z + µ κ1 ( d dz )−κ2 y1(z) ] , (6.2) satisfies the Fuchsian system DY (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃; k), where λ̃ and µ̃ are defined in equation (4.8). (iii) If κ2 ∈ Z<0 and y1(z) is a solution to Dy1(θ0, θ1, θt, θ∞;λ, µ), then the function ỹ(z) = ( d dz )−κ2 y1(z), satisfies Dy1(κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃). Proof. (i) If ν = −1, then it follows immediately that the function U = t(z−1y1(z), z−1y2(z), (z−1)−1y1(z), (z−1)−1y2(z), (z−t)−1y1(z), (z−t)−1y2(z)) satisfied equation (6.1) for n = 0. As- sume now that the function U = t(u1(z), u2(z), u3(z), u4(z), u5(z), u6(z)) satisfies equation (6.1). Set V = dU/dz. Since B (−1−n) 0 z2 + B (−1−n) 1 (z − 1)2 + B (−1−n) t (z − t)2 = 1 z B (−1−n) 0 z  u1(z) u2(z) 0 0 0 0 + 1 z − 1 B (−1−n) 0 z − 1  0 0 u3(z) u4(z) 0 0  + 1 z − t B (−1−n) 0 z − t  0 0 0 0 u5(z) u6(z)  = 1 z  u′1(z) u′2(z) 0 0 0 0 + 1 z − 1  0 0 u′3(z) u′4(z) 0 0 + 1 z − t  0 0 0 0 u′5(z) u′6(z)  , Middle Convolution and Heun’s Equation 15 we have dV dz = d dz {( B (−1−n) 0 z + B (−1−n) 1 z − 1 + B (−1−n) t z − t ) U } = − ( B (−1−n) 0 z2 + B (−1−n) 1 (z − 1)2 + B (−1−n) t (z − t)2 ) U + ( B (−1−n) 0 z + B (−1−n) 1 z − 1 + B (−1−n) t z − t ) V = ( B (−2−n) 0 z + B (−2−n) 1 z − 1 + B (−2−n) t z − t ) V. Hence (i) is proved inductively. (ii) Let A0, A1, At be the matrices defined by equation (2.5) and set ν = κ2. We define the matrix S by equation (4.4) and set Ũ = S−1U . Then Ỹ = t(ũ1(z), ũ2(z)) satisfies the Fuchsian differential equation DY (θ̃0, θ̃1, θ̃t, θ̃∞; λ̃, µ̃; k̃) where the parameters are determined by equation (4.5), and ũ1(z), ũ2(z) are expressed as equation (4.6). By a straightforward calculation as obtaining equation (4.7), we have equation (6.2). (iii) follows from (ii). � Note that (i) is valid for Fuchsian differential systems of arbitrary size and arbitrary number of regular singularities. We have a similar statement for the case ν = κ1 and κ1 ∈ Z<0. Namely, if κ1 ∈ Z<0 and Y = t(y1(z), y2(z)) is a solution to the Fuchsian differential equation DY (θ0, θ1, θt, θ∞;λ, µ; k) (see equation (2.9)), then the function Ỹ = t(ỹ1(z), ỹ2(z)) defined by ỹ1(z) = ( d dz )−κ1−1{ 1 z − λ ( κ1y1(z) + (z − λ̃) dy1(z) dz )} , ỹ2(z) = ( d dz )−κ1 y2(z), satisfies the Fuchsian system DY (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ; k). If κ2 = 0 (resp. κ1 = 0), then the Fuchsian system DY (κ2+θ0, κ2+θ1, κ2+θt, κ2+θ∞; λ̃, µ̃; k) (resp. DY (κ1 + θ0, κ1 + θ1, κ1 + θt,−κ2;λ + κ1/µ, µ; k)) coincides with DY (θ0, θ1, θt, θ∞;λ, µ; k), and the function Ỹ = t(ỹ1(z), ỹ2(z)) just corresponds to t(y1(z), y2(z)). On the case κ2 ∈ Z>0, we have the following proposition: Proposition 5. Let p ∈ {0, 1, t,∞} and Cp be the cycle starting from w = z, turning w = p anti-clockwise and return to w = z. (i) If κ2 ∈ Z>0 and Y = t(y1(z), y2(z)) is a solution to the Fuchsian system DY (θ0, θ1, θt, θ∞; λ, µ; k), then the function Ỹ = t(ỹ1(z), ỹ2(z)) defined by ỹ1(z) = ∫ Cp dy1(w) dw (z − w)κ2dw, (6.3) ỹ2(z) = κ2λ(λ− 1)(λ− t) k(λ− λ̃) ∫ Cp {( dy1(w) dw − µy1(w) ) 1 λ− w + µ κ1 dy1(w) dw } (z − w)κ2dw, satisfies the Fuchsian system DY (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃; k) for p ∈ {0, 1, t,∞} where λ̃ and µ̃ are defined in equation (4.8). (ii) If κ2 ∈ Z>0 and y1(z) is a solution to Dy1(θ0, θ1, θt, θ∞;λ, µ), then the function ỹ(z) = ∫ Cp y1(w)(z − w)κ2−1dw, (6.4) satisfies Dy1(κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃) for p ∈ {0, 1, t,∞}. 16 K. Takemura Note that the functions ỹ1(z), ỹ2(z), ỹ(z) may not be polynomials although the integrands of equations (6.3) and (6.4) are polynomials in z. Proof. Set K1(w) = dy1(w) dw , K2(w) = κ2λ(λ− 1)(λ− t) k(λ− λ̃) {( dy1(w) dw − µy1(w) ) 1 λ− w + µ κ1 dy1(w) dw } . It follows from Theorem 1 that the function Y (z) = ( ỹ1(z) ỹ2(z) ) defined by ỹi(z) = ∫ γzγpγ−1 z γ−1 p Ki(w)(z − w)κ2dw = (1− e2π √ −1κ2) ∫ γp Ki(w)(z−w)κ2dw + ∫ γz (Kγp i (w)−Ki(w))(z − w)κ2dw (i = 1, 2) is a solution to DY (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃; k) for p ∈ {0, 1, t}, where K γp i (w) is the function analytically continued along the cycle γp. If κ2 > −1, then the integrals ∫ γz (Kγp i (w) − Ki(w))(z − w)κ2dw tends to zero as the base point o of the integral tends to z, and it follows that the function ( ∫ Cp K1(w)(z − w)κ2dw∫ Cp K2(w)(z − w)κ2dw ) is a solution to DY (κ2 + θ0, κ2 + θ1, κ2 + θt, κ2 + θ∞; λ̃, µ̃; k) for κ2 ∈ R>−1 \Z>−1. By considering the limit κ2 → n, n ∈ Z>0, we obtain (i) for p ∈ {0, 1, t}. The case p = ∞ follows from C∞ = C−1 t C−1 1 C−1 0 . By integration by parts as equation (4.9) we obtain (ii). � We have similar proposition for the case ν = κ1 and κ1 ∈ Z>0. On middle convolution mcν for Fuchsian differential systems of arbitrary size and arbitrary number of regular singularities whose parameter ν is positive integer, the contour [γz, γp] can be replaced by Cp. We can reformulate Theorem 3 (resp. Theorem 4) for the case k2 ∈ Z<0 (resp. k1 ∈ Z<0) by changing the integral to successive differential and for the case k2 ∈ Z>0 (resp. k1 ∈ Z>0) by changing the contour [γz, γp] to Cp. The corresponding setting for Heun’s equation is described as follows: Proposition 6. Let v(w) be a solution to Heun’s equation written as equation (5.2) with the parameters in equation (5.1). (i) If η ∈ Z>1, then the function y(z) = (d/dz)η−1v(z) is a solution to Heun’s equation (5.3). (ii) If η ∈ Z<1, then the function y(z) = ∫ Cp v(w)(z − w)−ηdw is a solution to Heun’s equation (5.3) for p ∈ {0, 1, t,∞}. The generalized Darboux transformation (Crum–Darboux transformation) for elliptical rep- resentation of Heun’s equation was introduced in [29], and we can show that Proposition 6 (i) Middle Convolution and Heun’s Equation 17 gives another description of the generalized Darboux transformation. Hence the integral trans- formation given by Theorem 2 can be regarded as a generalization of the generalized Darboux transformation to non-integer cases. Khare and Sukhatme [15] conjectured a duality of quasi- exactly solvable (QES) eigenvalues for elliptical representation of Heun’s equation. By rewriting parameters of the duality to Heun’s equation on the Riemann sphere, we obtain a correspon- dence on the parameters α, β, γ, δ, ε, q and α′, β′, γ′, δ′, ε′, q′ on the integral transformation of Heun’s equation in Theorem 2. We will report further from a viewpoint of monodromy in a separated paper. A Appendix We investigate the realization of the Fuchsian system (equation (2.1)) for the cases λ = 0, 1, t,∞ in the setting of Section 2 and observe relationships with the lines L0, L∗0, L1, L∗1, Lt, L∗t , L∞, L∗∞ (see equation (3.3)) in the space of initial conditions E(t). We consider the case λ = 0, i.e., the case a (0) 12 = 0, a (1) 12 6= 0, a (t) 12 6= 0, (t+1)a(0) 12 +ta (1) 12 +a (t) 12 6= 0 (see equation (2.4)). Since a (0) 12 = 0 and the eigenvalues of A0 are θ0 and 0, the matrix A0 is written as A0 = ( θ0 0 v 0 ) or A0 = ( 0 0 v θ0 ) , and it follows from a (1) 12 6= 0, a (t) 12 6= 0 that the matrices A1, At may be expressed as equation (2.5). We determine w1, wt so as to satisfy a12(z) = −w1(z− 1)−wt/(z− t) = k/(z(z− 1)). Then we have w1 = k/(t− 1), wt = −k/(t− 1). (A.1) On the case A0 = ( 0 0 v θ0 ) , A1 = ( u1 + θ1 −w1 u1(u1 + θ1)/w1 −u1 ) , At = ( ut + θt −wt ut(ut + θt)/wt −ut ) , (A.2) we determine u1, ut so as to satisfy equation (2.2), namely v + u1(u1 + θ1)/w1 + ut(ut + θt)/wt = 0, u1 + θ1 + ut + θt = −κ1, θ0 − u1 − ut = −κ2. We have u1 = −θ1 + 1 θ∞ − θ0 ( kv t− 1 − κ1(κ1 + θt) ) , ut = −θt − 1 θ∞ − θ0 ( kv t− 1 + κ1(κ1 + θ1) ) . (A.3) Hence we have one-parameter realization of equation (A.2) with the prescribed condition. We discuss relationship with the Fuchsian system on the line L0. For this purpose, we recall matri- ces A0, A1, At determined by equations (2.5), (2.7), (2.8) and restrict them to λ = 0. Then all elements in A0, A1 and At are well-defined and we have A0 \|λ=0 = ( 0 0 θ0{t(θ0 − θ∞)µ + κ1(κ1 + θ1 + t(κ1 + θt))}/(kθ∞) θ0 ) . 18 K. Takemura In fact the matrices A0, A1, At restricted to the line L0 coincide with the ones determined by equations (A.2), (A.1), (A.3) and v = θ0{t(θ0 − θ∞)µ + κ1(κ1 + θ1 + t(κ1 + θt))}/(kθ∞). Note that the second-order differential equation for the function y1 on the case of the matrices in equations (A.2), (A.1), (A.3) is obtained as equation (3.4) by substituting µ = (kθ∞v − κ1θ0(κ1 + θ1 + t(κ1 + θt)))/(tθ0(θ0 − θ∞)). On the case A0 = ( θ0 0 v 0 ) , A1 = ( u1 + θ1 −w1 u1(u1 + θ1)/w1 −u1 ) , At = ( ut + θt −wt ut(ut + θt)/wt −ut ) , (A.4) u1, ut are determined as u1 = 1 θ∞ + θ0 ( kv t− 1 − κ2(κ2 + θt) ) , ut = −1 θ∞ + θ0 ( kv t− 1 + κ2(κ2 + θ1) ) , (A.5) to satisfy equation (2.2). To realize the Fuchsian system on the line L∗0, we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8), transform (λ, µ)(= (q0, p0)) to (q1, p1) by equation (3.2) and restrict matrix elements to q1 = 0. Then the matrices A0, A1 and At are determined as equations (A.4), (A.1), (A.5), where v = {−t(θ0 + θ∞)q1 + θ0(κ1 + θ0)((κ2 + θt) + t(κ2 + θ1))}/(kθ∞). (A.6) Note that the second-order differential equation for the function ỹ1 = z−1y1 on the case of the matrices in equations (A.4), (A.1), (A.5) is obtained as equation (3.8) by substituting equa- tion (A.6). We consider the case λ = 1, i.e., the case a (1) 12 = 0, a (0) 12 6= 0, a (t) 12 6= 0, (t+1)a(0) 12 +ta (1) 12 +a (t) 12 6= 0. Since a (1) 12 = 0 and the eigenvalues of A1 are θ1 and 0, the matrix A1 is written as A1 = ( θ1 0 v 0 ) or A1 = ( 0 0 v θ1 ) , and the matrices A0, At may be expressed as equation (2.5). To satisfy a12(z) = −w0/z − wt/(z − t) = k/(z(z − t)), we have w0 = k/t, wt = −k/t. (A.7) On the case A1 = ( 0 0 v θ1 ) , A0 = ( u0 + θ0 −w0 u0(u0 + θ0)/w0 −u0 ) , At = ( ut + θt −wt ut(ut + θt)/wt −ut ) , (A.8) u0, ut are determined as u0 = −θ0 + 1 θ∞ − θ1 ( kv t − κ1(κ1 + θt) ) , ut = −θt − 1 θ∞ − θ1 ( kv t + κ1(κ1 + θ0) ) , (A.9) Middle Convolution and Heun’s Equation 19 to satisfy equation (2.2). To realize the Fuchsian system on the line L1, we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8) and restrict matrix elements to λ = 1. Then the matrices A0, A1 and At are determined as equations (A.8), (A.7), (A.9), where v = θ1{(1− t)(θ1 − θ∞)µ− κ1(κ1 + θ0 + (1− t)(κ1 + θt))}/(kθ∞). (A.10) Note that the second-order differential equation for the function y1 on the case of the matrices in equations (A.8), (A.7), (A.9) is obtained as equation (3.5) by substituting equation (A.10). On the case A1 = ( θ1 0 v 0 ) , A0 = ( u0 + θ0 −w0 u0(u0 + θ0)/w0 −u0 ) , At = ( ut + θt −wt ut(ut + θt)/wt −ut ) , (A.11) u0, ut are determined as u0 = 1 θ∞ + θ1 ( kv t − κ2(κ2 + θt) ) , ut = −1 θ∞ + θ1 ( kv t + κ2(κ2 + θ0) ) , (A.12) to satisfy equation (2.2). To realize the Fuchsian system on the line L∗1, we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8), transform (λ, µ)(= (q0, p0)) to (q2, p2) by equation (3.2) and restrict matrix elements to q2 = 0. Then the matrices A0, A1 and At are determined as equations (A.11), (A.7), (A.12), where v = {(t− 1)(θ1 + θ∞)q2 − θ1(κ1 + θ1)((κ2 + θt) + (1− t)(κ2 + θ0))}/(kθ∞). (A.13) Note that the second-order differential equation for the function ỹ1 = (z − 1)−1y1 on the case of the matrices in equations (A.11), (A.7), (A.12) is obtained as equation (3.9) by substituting equation (A.13). We consider the case λ = t, i.e., the case a (t) 12 = 0, a (0) 12 6= 0, a (1) 12 6= 0, (t+1)a(0) 12 +ta (1) 12 +a (t) 12 6= 0. Then the matrix At is written as At = ( θt 0 v 0 ) or At = ( 0 0 v θt ) , and the matrices A0, A1 may be expressed as equation (2.5). To satisfy a12(z) = −w0/z − w1/(z − 1) = k/(z(z − 1)), we have w0 = k, w1 = −k. (A.14) On the case At = ( 0 0 v θt ) , A0 = ( u0 + θ0 −w0 u0(u0 + θ0)/w0 −u0 ) , A1 = ( u1 + θ1 −w1 u1(u1 + θ1)/w1 −u1 ) , (A.15) u0, u1 are determined as u0 = −θ0 + 1 θ∞ − θt (kv − κ1(κ1 + θ1)) , u1 = −θ1 − 1 θ∞ − θt (kv + κ1(κ1 + θ0)) , (A.16) 20 K. Takemura to satisfy equation (2.2). To realize the Fuchsian system on the line Lt, we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8) and restrict matrix elements to λ = t. Then the matrices A0, A1 and At are determined as equations (A.15), (A.14), (A.16), where v = θt{t(t− 1)(θt − θ∞)µ− κ1(t(κ1 + θ0) + (t− 1)(κ1 + θ1))}/(kθ∞). (A.17) Note that the second-order differential equation for the function y1 on the case of the matrices in equations (A.15), (A.14), (A.16) is obtained as equation (3.6) by substituting equation (A.17). On the case At = ( θt 0 v 0 ) , A0 = ( u0 + θ0 −w0 u0(u0 + θ0)/w0 −u0 ) , A1 = ( u1 + θ1 −w1 u1(u1 + θ1)/w1 −u1 ) , (A.18) u0, ut are determined as u0 = 1 θ∞ + θt (kv − κ2(κ2 + θ1)) , u1 = −1 θ∞ + θt (kv + κ2(κ2 + θ0)) , (A.19) to satisfy equation (2.2). To realize the Fuchsian system on the line L∗t , we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8), transform (λ, µ)(= (q0, p0)) to (q3, p3) by equation (3.2) and restrict matrix elements to q3 = 0. Then the matrices A0, A1 and At are determined as equations (A.18), (A.14), (A.19), where v = {t(1− t)(θt + θ∞)q3 − θt(κ1 + θt)(t(κ2 + θ1) + (t− 1)(κ2 + θ0))}/(kθ∞). (A.20) Note that the second-order differential equation for the function ỹ1 = (z− 1)−1y1 on the case of the matrices in equations (A.18), (A.14), (A.19) is obtained as equation (3.10) by substituting equation (A.20). We consider the case λ = ∞, i.e., the case a (0) 12 6= 0, a (1) 12 6= 0, a (t) 12 6= 0, (t + 1)a(0) 12 + ta (1) 12 + a (t) 12 = 0. We can set A0, A1, At as equation (2.5) and we determine u0, u1, ut, w0, w1, wt to satisfy equation (2.2) and a12(z) = −w0 z − w1 z − 1 − wt z − t = k z(z − 1)(z − t) . Then we have w0 = −k/t, w1 = k/(t− 1), wt = −k/(t(t− 1)), (A.21) and other relations are written as u0(u0 + θ0)/w0 + u1(u1 + θ1)/w1 + ut(ut + θt)/wt = 0, u0 + θ0 + u1 + θ1 + ut + θt = −κ1, −u0 − u1 − ut = −κ2. We solve the equation for u0, u1, ut by adding one more relation −(u1 + θ1 + t(ut + θt)) = l. We have u0 = −θ0 − κ1 + l̃ tθ∞ , u1 = −θ1 + l̃ − θ∞l (1− t)θ∞ , ut = −θt + l̃ − tθ∞l t(t− 1)θ∞ , l̃ = l2 + (θ1 + tθt)l + tκ1(κ1 + θ0). (A.22) Middle Convolution and Heun’s Equation 21 The second-order differential equation for the function y1 is written as d2y1 dz2 + ( 1− θ0 z + 1− θ1 z − 1 + 1− θt z − t ) dy1 dz + κ1(κ2 + 2)z − q z(z − 1)(z − t) y1 = 0, q = l(θ∞ − 1) + κ1(t(κ2 + θt + 1) + κ2 + θ1 + 1). To realize the Fuchsian system on the line L∞, we recall matrices A0, A1, At determined by equations (2.5), (2.7), (2.8), transform (λ, µ)(= (q0, p0)) to (q∞, p∞) by equation (3.2), replace k by −kq∞ and restrict matrix elements to q∞ = 0. Then the matrices A0, A1 and At are determined as equations (2.5), (A.21), (A.22), where l = p∞. We have observed that Fuchsian systems on the lines L0, L∗0, L1, L∗1, Lt, L∗t , L∞ are realized by Fuchsian systems for the case λ = 0, 1, t,∞. Here the case L∗∞ is missing. In fact this case does not simply correspond with the Fuchsian system as equation (2.1), because we have u0 = 1 tθ∞ 1 p2 4 + O ( p−1 4 ) , u1 = 1 (1− t)θ∞ 1 p2 4 + O ( p−1 4 ) , ut = 1 t(t− 1)θ∞ 1 p2 4 + O ( p−1 4 ) , as p4 → 0 in the coordinate (q4, p4) in equation (3.2), although we can restrict the second-order differential equation for y1 to p4 = 0 as equation (3.11). Acknowledgements The author would like to thank Alexander Kazakov for sending the paper [14] with valuable comments. The author is supported by the Grant-in-Aid for Young Scientists (B) (No. 19740089) from the Japan Society for the Promotion of Science. References [1] Boalch P., From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3) 90 (2005), 167–208, math.AG/0308221. [2] Dettweiler M., Reiter S., An algorithm of Katz and its application to the inverse Galois problem. Algorithmic methods in Galois theory, J. Symbolic Comput. 30 (2000), 761–798. 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(4) 27 (1998), 379–425. http://arxiv.org/abs/quant-ph/0602105 http://arxiv.org/abs/math.CA/0109149 http://arxiv.org/abs/math.CA/0103077 http://arxiv.org/abs/math.CA/0112179 http://arxiv.org/abs/math.CA/0201208 http://arxiv.org/abs/math.CA/0406141 http://arxiv.org/abs/math.CA/0508093 http://arxiv.org/abs/0705.3358 http://arxiv.org/abs/0810.2766 1 Introduction 2 Fuchsian system of rank two with four singularities 3 The space of initial conditions for the sixth Painlevé equation and Heun's equation 4 Middle convolution 5 Middle convolution, integral transformations of Heun's equation and the space of initial conditions 6 Middle convolution for the case that the parameter is integer A Appendix References

Middle Convolution and Heun's Equation

Institution

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