Abstract
In our previous works, a relationship between Hermite’s two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study \({\tau}\)-functions associated with holonomic deformations of linear differential equations by using Hermite’s two approximation problems. As a result, we present a determinant formula for the ratio of \({\tau}\)-functions (\({\tau}\)-quotient).
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Chudnovsky D.V., Chudnovsky G.V.: Bäcklund transformations for linear differential equations and Padé approximations. I. J. Math. Pures Appl. 61, 1–16 (1982)
Chudnovsky D.V., Chudnovsky G.V.: Explicit continued fractions and quantum gravity. Acta Appl. Math. 36, 167–185 (1994)
Ishikawa M., Okada S.: Identities for determinants and Pfaffians, and their applications. Sugaku Expos. 27, 85–116 (2014)
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)
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)
Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: A Modern Theory of Special Functions. Vieweg, Braunschweig (1991)
Jimbo M., Miwa T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2, 407–448 (1981)
Jimbo M., Miwa T., Ueno K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. Phys. D 2, 306–352 (1981)
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)
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)
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)
Knuth, D.: Overlapping Pfaffians. Electron. J. Combin. 3, no. 2, #R5 (1996)
Magnus A.: Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. 57, 215–237 (1995)
Mahler K.: Perfect systems. Compos. Math. 19, 95–166 (1968)
Mano T.: Determinant formula for solutions of the Garnier system and Padé approximation. J. Phys. A 45, 135206 (2012)
Mano, T., Tsuda, T.: Two approximation problems by Hermite, and Schlesinger transformation. RIMS Kokyuroku Bessatsu B 47, 77–86 (2014) (in Japanese)
Mano T., Tsuda T.: Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral. Math. Z. 285, 397–431 (2017)
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)
Matsumoto S.: Hyperdeterminantal expressions for Jack functions of rectangular shapes. J. Algebra 320, 612–632 (2008)
Ohta, Y.: Bilinear Theory of Solitons. Doctoral Thesis, Graduate School of Engineering, University of Tokyo (1992)
Oshima T.: Classification of Fuchsian systems and their connection problem. RIMS Kokyuroku Bessatsu B 37, 163–192 (2013)
Suzuki T.: Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems. J. Math. Phys. 55, 102902 (2014)
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)
Tsuda T.: Rational solutions of the Garnier system in terms of Schur polynomials. Int. Math. Res. Not. 43, 2341–2358 (2003)
Tsuda T.: Toda equation and special polynomials associated with the Garnier system. Adv. Math. 206, 657–683 (2006)
Tsuda T.: UC hierarchy and monodromy preserving deformation. J. Reine Angew. Math. 690, 1–34 (2014)
Wenzel W.: Pfaffian forms and \({\Delta}\)-matroids. Discrete Math. 115, 253–266 (1993)
Yamada Y.: Padé method to Painlevé equations. Funkcial. Ekvac. 52, 83–92 (2009)
Acknowledgements
This work was supported by a grant-in-aid from the Japan Society for the Promotion of Science (Grant Numbers 16K05068, 17K05270, 17K05335, 25800082 and 25870234).
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Ishikawa, M., Mano, T. & Tsuda, T. Determinant Structure for \({\tau}\)-Function of Holonomic Deformation of Linear Differential Equations. Commun. Math. Phys. 363, 1081–1101 (2018). https://doi.org/10.1007/s00220-018-3256-z
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DOI: https://doi.org/10.1007/s00220-018-3256-z