Schlesinger’s Equations for Upper Triangular Matrices and Their Solutions

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We consider explicit integral expressions of hypergeometric and hyperelliptic types for solutions of Schlesinger’s equations in classes of upper triangular matrices with eigenvalues that produce arithmetic progressions with the same difference. These integral representations extend and generalize earlier known results.

About the authors

V P Lexin

State Socio-Humanitarian University

Kolomna, Russia


  1. Болибрух А. А. Обратные задачи монодромии в аналитической теории дифференциальных уравнений. - М.: МЦНМО, 2009.
  2. Уиттекер Э. Т., Ватсон Дж. Н. Курс современного анализа. - М.: Физматлит, 1963.
  3. Aomoto K. On the structure of integrals of power product of linear functions// Sci. Papers College Gen. Edu. Univ. Tokyo. - 1977. - 27, № 2. - С. 49-61.
  4. Aomoto K. Founctions hyperlogarithmiques et groupes de monodromie unipotens// Sci. J. Fac. Sci. Univ. Tokyo. - 1978. - 25. - С. 149-156.
  5. Deligne P., Mostow G. D. Monodromy of hypergeometric functions and non-lattice integral monodromy// Publ. IHES. - 1986. - 63. - С. 5-90.
  6. Dragovich V., Schramchenko V. Algebro-geometric solutions to triangular Schlesinger systems// arxiv: 1604.01820v2[math.AG].
  7. Dubrovin B., Mazzocco M. On the reductions and classical solutions of the Schlesinger equations// В сб. «Differential Equations and Quantum Groups». - Strasburg: IRMA, 2007. - C. 157-187.
  8. Gontsov R. R., Leksin V. P. On the reducibility of Schlesinger isomonodromic families// В сб. «Analytic Methods of Analysis and Differential Equations: AMADE 2012». - Cambridge: Cambridge Sci. Publ., 2014. - C. 21-34.
  9. Kapovich M., Millson J. Quantization of bending deformations of polygons in E3, hypergeometric integrals and the Gassner representation// Can. Math. Bull. - 2001. - 44, № 1. - C. 36-60.
  10. Katz N., Oda T. On the differentiation of de Rham cohomology classes with respect to parameters// J. Math. Kyoto Univ. - 1968. - 8. - C. 199-213.
  11. Kohno T. Linear representations of braid groups and classical Yang-Baxter equations// Contemp. Math. - 1988. - 78. - C. 339-363.
  12. Leksin V. P. Isomonodromy deformations and hypergeometric-type systems// В сб. «Painleve´ Equations and Related Topics». - Berlin-Boston: Walther de Gruyter, 2012. - C. 117-122.
  13. Zˇoladek H. The Monodromy Group. - Basel-Boston-Berlin: Birkha¨user, 2006.



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