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A STUDY OF A FUCHSIAN SYSTEM OF RANK 8 IN 3 VARIABLES AND THE ORDINARY DIFFERENTIAL EQUATIONS AS ITS RESTRICTIONS

Ebisu, Akihito 大阪大学 DOI:10.18910/89997

2023.01

概要

A Fuchsian system of rank 8 in 3 variables with 4 parameters is found. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equations, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions of these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series ₄F₃ at 1.

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参考文献

[1] G.E. Andrews, R. Askey and R. Roy: Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.

[2] V.S. Dotsenko and V.A. Fateev: Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Phys. B 240 (1984), 312–348.

[3] A. Ebisu: Invariants of difference equations and transformation formulae for hypergeometric functions, in preparation.

[4] A. Ebisu and H. Ochiai: Difference equations for 4F3(1), in preparation.

[5] Y. Haraoka: Prolongability of ordinary differential equations, J. Nonlinear. Math. Physics 20, Supplement 1, 70–84.

[6] Y. Haraoka: On Oshima’s middle convolution, Josai Math. Monographs 12 (2020), 19–51.

[7] E.J. Wilczynski: Projective Differential Geometry of Curves and Ruled Surfaces, reprinted by Chelsea Pub. Co., Teubner, 1906.

[8] J.A. Wilson: Three-term contiguous relations and some new orthogonal polynomials: in Pade and rational ´ approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976), 227–232, Academic Press, New York, 1977.

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