Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

Novikov, D. P.

doi:10.1007/s11232-006-0040-6

Cited by 15 publications

(16 citation statements)

References 12 publications

(19 reference statements)

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“…The absence of the variable u in the first equation allows simplifying the proof of Theorem 2 in [15]. Equation (3) was found similarly, by eliminating the variables A i from system (2) based on the theory in [16].…”

Section: Systems Of Linear Equations Accompanying the Painlevé And Scmentioning

confidence: 99%

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Novikov

2009

Theor Math Phys

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We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 matrices. This generalizes Suleimanov's result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function. Systems of linear equations accompanying the Painlevé andSchlesinger equations The Schlesinger system [1] for the matriceswhere i, j = 1, m (the second group of equations can be replaced with the condition that the matrix A 1 + · · · + A m = A ∞ is constant in t 1 , . . . , t m ), was discovered as the compatibility condition for the system(2)All algebraic integrals of motion of system (1) are known. They are the traces tr A k i , k = 1, 2, . . . (equivalently, the characteristic polynomials of the matrices A i ). A closed form ω = H i dt i was found in [2],and the τ -function (log τ ) ti = H i was also determined.Here, we consider only one aspect of the problem of the complete integrability of system (1) in terms of special functions. Namely, in the case of 2×2 matrices A i , we seek a second-order linear equation for

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Section: Systems Of Linear Equations Accompanying the Painlevé And Scmentioning

confidence: 99%

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Novikov

2009

Theor Math Phys

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“…Since the middle convolution is realized by the Riemann-Liouville integral (3.8), the above process can be regarded as a way of obtaining Bäcklund transformations by using an integral transformation, which is done by Novikov in [11].…”

Section: Note Thatmentioning

confidence: 99%

“…It is also well known that this equation possesses the so-called Painlevé property: every solution may be analytically continued to a meromorphic function on the universal cover of CP 1 \ {0, 1, ∞}. Relations to the works by Boalch [1,2], Harnad [6], Mazzocco [10] and Novikov [11] are also clarified in Section 5.…”

Section: Introductionmentioning

confidence: 96%

Middle convolution and deformation for Fuchsian systems

Haraoka

Filipuk²

2007

Journal of the London Mathematical Society

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“…It would be interesting to prove that the conditions (1.5)-(1.7) and (7.2) imply uniqueness. We finally note that kernel functions have been used since a long time to transform the Heun equation into integral equations [3,18]; see also [13,31]. This has provided powerful tools to study analytical properties of solutions; for example, this was used by Ruijsenaars in his work on the hidden permutation symmetry mentioned after (5.10) [35].…”

Section: Final Remarksmentioning

confidence: 99%

Series Solutions of the Non-Stationary Heun Equation

Atai¹,

Langmann²

2018

SIGMA

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Abstract. We consider the non-stationary Heun equation, also known as quantum Painlevé VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the problem to solve this equation into a differential-difference equation which, as we show, can be solved by efficient recursive algorithms. We thus obtain series representations of solutions which provide elliptic generalizations of the Jacobi polynomials. These series reproduce, in a limiting case, a perturbative solution of the Heun equation due to Takemura, but our method is different in that we expand in non-conventional basis functions that allow us to obtain explicit formulas to all orders; in particular, for special parameter values, our series reduce to a single term.

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Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

Cited by 15 publications

References 12 publications

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

Middle convolution and deformation for Fuchsian systems

Series Solutions of the Non-Stationary Heun Equation

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