Determinant Structure for \({\tau}\)-Function of Holonomic Deformation of Linear Differential Equations

1. Chudnovsky D.V., Chudnovsky G.V.: Bäcklund transformations for linear differential equations and Padé approximations. I. J. Math. Pures Appl. 61, 1–16 (1982)

   MATH  Google Scholar 

2. Chudnovsky D.V., Chudnovsky G.V.: Explicit continued fractions and quantum gravity. Acta Appl. Math. 36, 167–185 (1994)

   Article  MathSciNet  Google Scholar 

3. Ishikawa M., Okada S.: Identities for determinants and Pfaffians, and their applications. Sugaku Expos. 27, 85–116 (2014)

   Google Scholar 

4. Ishikawa M., Wakayama M.: Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities. J. Combin. Theory Ser. A 113, 136–157 (2006)

   Article  Google Scholar 

5. Iwasaki K., Kajiwara K., Nakamura T.: Generating function associated with the rational solutions of the Painlevé II equation. J. Phys. A 35, L207–L211 (2002)

   Article  ADS  Google Scholar 

6. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: A Modern Theory of Special Functions. Vieweg, Braunschweig (1991)

   Chapter  Google Scholar 

7. Jimbo M., Miwa T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2, 407–448 (1981)

   Article  MathSciNet  Google Scholar 

8. Jimbo M., Miwa T., Ueno K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. Phys. D 2, 306–352 (1981)

   Article  MathSciNet  Google Scholar 

9. Joshi N., Kajiwara K., Mazzocco M.: Generating function associated with the determinant formula for the solutions of the Painlevé II equation. Astérisque 297, 67–78 (2004)

   MATH  Google Scholar 

10. Joshi N., Kajiwara K., Mazzocco M.: Generating function associated with the Hankel determinant formula for the solutions of the Painlevé IV equation. Funkcial. Ekvac. 49, 451–468 (2006)

    Article  MathSciNet  Google Scholar 

11. Kajiwara K., Mazzocco M., Ohta Y.: A remark on the Hankel determinant formula for solutions of the Toda equation. J. Phys. A 40, 12661–12675 (2007)

    Article  ADS  MathSciNet  Google Scholar 

12. Knuth, D.: Overlapping Pfaffians. Electron. J. Combin. 3, no. 2, #R5 (1996)

13. Magnus A.: Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57, 215–237 (1995)

    Article  MathSciNet  Google Scholar 

14. Mahler K.: Perfect systems. Compos. Math. 19, 95–166 (1968)

    MathSciNet  MATH  Google Scholar 

15. Mano T.: Determinant formula for solutions of the Garnier system and Padé approximation. J. Phys. A 45, 135206 (2012)

    Article  ADS  MathSciNet  Google Scholar 

16. Mano, T., Tsuda, T.: Two approximation problems by Hermite, and Schlesinger transformation. RIMS Kokyuroku Bessatsu B 47, 77–86 (2014) (in Japanese)

17. Mano T., Tsuda T.: Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral. Math. Z. 285, 397–431 (2017)

    Article  MathSciNet  Google Scholar 

18. Masuda T.: On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46, 121–171 (2003)

    Article  MathSciNet  Google Scholar 

19. Matsumoto S.: Hyperdeterminantal expressions for Jack functions of rectangular shapes. J. Algebra 320, 612–632 (2008)

    Article  MathSciNet  Google Scholar 

20. Ohta, Y.: Bilinear Theory of Solitons. Doctoral Thesis, Graduate School of Engineering, University of Tokyo (1992)

21. Oshima T.: Classification of Fuchsian systems and their connection problem. RIMS Kokyuroku Bessatsu B 37, 163–192 (2013)

    MathSciNet  MATH  Google Scholar 

22. Suzuki T.: Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems. J. Math. Phys. 55, 102902 (2014)

    Article  ADS  MathSciNet  Google Scholar 

23. Tsuda T.: Birational symmetries, Hirota bilinear forms and special solutions of the Garnier systems in 2-variables. J. Math. Sci. Univ. Tokyo 10, 355–371 (2003)

    MathSciNet  MATH  Google Scholar 

24. Tsuda T.: Rational solutions of the Garnier system in terms of Schur polynomials. Int. Math. Res. Not. 43, 2341–2358 (2003)

    Article  MathSciNet  Google Scholar 

25. Tsuda T.: Toda equation and special polynomials associated with the Garnier system. Adv. Math. 206, 657–683 (2006)

    Article  MathSciNet  Google Scholar 

26. Tsuda T.: UC hierarchy and monodromy preserving deformation. J. Reine Angew. Math. 690, 1–34 (2014)

    Article  MathSciNet  Google Scholar 

27. Wenzel W.: Pfaffian forms and \({\Delta}\)-matroids. Discrete Math. 115, 253–266 (1993)

    Article  MathSciNet  Google Scholar 

28. Yamada Y.: Padé method to Painlevé equations. Funkcial. Ekvac. 52, 83–92 (2009)

    Article  MathSciNet  Google Scholar 

Download references