We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.
@article{bwmeta1.element.doi-10_2478_s11533-008-0059-z,
author = {Lev Sakhnovich},
title = {On the solutions of Knizhnik-Zamolodchikov system},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {145-162},
zbl = {1179.35276},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0059-z}
}
Lev Sakhnovich. On the solutions of Knizhnik-Zamolodchikov system. Open Mathematics, Tome 7 (2009) pp. 145-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0059-z/
[1] Bateman H., Erdely A., Higher transcendental functions 1, New York, 1953
[2] Burrow M., Representation theory of finite groups, Academic Press, New York-London, 1965
[3] Etingof P.I., Frenkel I.B., Kirillov A.A.(Jr.), Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs 58, American Mathematical Society, Providence, RI, 1998 | Zbl 0903.17006
[4] Felder G., Veselov A., Polynomial solutions of the Knizhnik-Zamolodchikov equations and Schur-Weyl duality, Int. Math. Res. Not. IMRN, 2007, 15, 1–27 | Zbl 1130.35024
[5] Matsuo A., An application of Aomoto-Gelfand hypergeometric functions to the SU(n) Knizhnik-Zamolodchikov equation. Comm. Math. Phys., 1990, 134, 65–77 http://dx.doi.org/10.1007/BF02102089[Crossref] | Zbl 0714.33012
[6] Sakhnovich L.A., Spectral theory of canonical differential systems. Method of operator identities, Operator Theory: Advances and Applications 107, Birkhäuser Verlag, Basel, 1999 | Zbl 0918.47003
[7] Sakhnovich L.A., Meromorphic solutions of linear differential systems, Painleve type functions, Oper. Matrices, 2007, 1, 87–111 | Zbl 1114.34068
[8] Sakhnovich L.A., Explicit rational solutions of Knizhnik-Zamolodchikov equation, Cent. Eur. J. Math., 2008, 6, 179–187 http://dx.doi.org/10.2478/s11533-008-0013-0[Crossref] | Zbl 1153.34054
[9] Sakhnovich L.A., Rational solutions of KZ equation, Existence and construction, preprint available at arXiv.math:- ph/0609067
[10] Sakhnovich L.A., Rational solutions of KZ equation, Case S4, preprint available at arXiv:math.CA/0702404
[11] Tydnyuk A., Rational solution of KZ equation, preprint available at arXiv:math/0162153
[12] Tydnyuk A., Explicit rational solution of KZ equation, preprint available at arXiv:math/07091141
[13] Varchenko A., Asymptotic solutions of Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys., 1995, 171, 99–137 http://dx.doi.org/10.1007/BF02103772[Crossref] | Zbl 0893.17009