@article{SIGMA_2010_6_a83,
author = {Nobutaka Nakazono},
title = {Hypergeometric $\tau$ {Functions} of the $q${-Painlev\'e} {Systems} of {Type} $(A_2+A_1)^{(1)}$},
journal = {Symmetry, integrability and geometry: methods and applications},
year = {2010},
volume = {6},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a83/}
}
Nobutaka Nakazono. Hypergeometric $\tau$ Functions of the $q$-Painlevé Systems of Type $(A_2+A_1)^{(1)}$. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a83/
[1] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl
[2] Hamamoto T., Kajiwara K., “Hypergeometric solutions to the $q$-Painlevé equation of type $A_4^{(1)}$”, J. Phys. A: Math. Theor., 40 (2007), 12509–12524, arXiv: nlin.SI/0701001 | DOI | MR | Zbl
[3] Hamamoto T., Kajiwara K., Witte N. S., “Hypergeometric solutions to the $q$-Painlevé equation of type $(A_1+A'_1)^{(1)}$”, Int. Math. Res. Not., 2006 (2006), Art. ID 84619, 26 pp., arXiv: nlin.SI/0607065 | DOI | MR
[4] 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 (2006), 451–468, arXiv: nlin.SI/0512041 | DOI | MR | Zbl
[5] Jimbo M., Miwa T., Ueno K., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function”, Phys. D, 2 (1981), 306–352 | DOI | MR | Zbl
[6] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D, 2 (1981), 407–448 | DOI | MR | Zbl
[7] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III”, Phys. D, 4 (1981/82), 26–46 | DOI | MR | Zbl
[8] Kajiwara K., Kimura K., “On a $q$-difference Painlevé III equation. I. Derivation, symmetry and Riccati type solutions”, J. Nonlin. Math. Phys., 10 (2003), 86–102, arXiv: nlin.SI/0205019 | DOI | MR | Zbl
[9] Kajiwara K., Masuda T., “A generalization of determinant formulae for the solutions of Painlevé II and XXXIV equations”, J. Phys. A: Math. Gen., 32 (1999), 3763–3778, arXiv: solv-int/9903014 | DOI | MR | Zbl
[10] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “${}_{10}E_9$ solution to the elliptic Painlevé equation”, J. Phys. A: Math. Gen., 36 (2003), L263–L272, arXiv: nlin.SI/0303032 | DOI | MR | Zbl
[11] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “Point configurations, Cremona transformations and the elliptic difference Painlevé equation”, Théories Asymptotiques et Équations de Painlevé, Semin. Congr., 14, Soc. Math. France, Paris, 2006, 169–198, arXiv: nlin.SI/0411003 | MR | Zbl
[12] Kajiwara K., Nakazono N., Tsuda T., “Projective reduction of the discrete Painlevé system of type $(A_2+A_1)^{(1)}$”, Int. Math. Res. Not., 2010 (2010), Art. ID rnq089, 37 pp., arXiv: 0910.4439 | DOI
[13] Kajiwara K., Noumi M., Yamada Y., “A study on the fourth $q$-Painlevé equation”, J. Phys. A: Math. Gen., 34 (2001), 8563–8581, arXiv: nlin.SI/0012063 | DOI | MR | Zbl
[14] Kajiwara K., Ohta Y., “Determinant structure of the rational solutions for the Painlevé IV equation”, J. Phys. A: Math. Gen., 31 (1998), 2431–2446, arXiv: solv-int/9709011 | DOI | MR | Zbl
[15] Kajiwara K., Ohta Y., Satsuma J., “Casorati determinant solutions for the discrete Painlevé III equation”, J. Math. Phys., 36 (1995), 4162–4174, arXiv: solv-int/9412004 | DOI | MR | Zbl
[16] Kajiwara K., Ohta Y., Satsuma J., Grammaticos B., Ramani A., “Casorati determinant solutions for the discrete Painlevé-II equation”, J. Phys. A: Math. Gen., 27 (1994), 915–922, arXiv: solv-int/9310002 | DOI | MR | Zbl
[17] Masuda T., “Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E_7^{(1)}$”, SIGMA, 5 (2009), 035, 30 pp., arXiv: 0903.4102 | DOI | MR | Zbl
[18] Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E_8^{(1)}$, MI Preprint Series, 2009–12, Kyushu University, 2009
[19] Miwa T., Jimbo M., Date E., Solitons. Differential equations, symmetries and infinite-dimensional algebras, Cambridge Tracts in Mathematics, 135, Cambridge University Press, Cambridge, 2000 | MR | Zbl
[20] Nakao S., Kajiwara K., Takahashi D., Multiplicative dP$_{\rm II}$ and its ultradiscretization, Reports of RIAM Symposium, No. 9ME-S2, Kyushu University, 1998, 125–130 (in Japanese)
[21] Noumi M., Painlevé equations through symmetry, Translations of Mathematical Monographs, 223, American Mathematical Society, Providence, RI, 2004 | MR | Zbl
[22] Noumi M., Yamada Y., “Symmetries in the fourth Painlevé equation and Okamoto polynomials”, Nagoya Math. J., 153 (1999), 53–86, arXiv: q-alg/9708018 | MR | Zbl
[23] Ohta Y., Nakamura A., “Similarity KP equation and various different representations of its solutions”, J. Phys. Soc. Japan, 61 (1992), 4295–4313 | DOI | MR
[24] Okamoto K., “Studies on the Painlevé equations. I. Sixth Painlevé equation P$_{\rm VI}$”, Ann. Mat. Pura Appl., 146 (1987), 337–381 | DOI | MR | Zbl
[25] Okamoto K., “Studies on the Painlevé equations. II. Fifth Painlevé equation P$_{\rm V}$”, Japan. J. Math. (N.S.), 13 (1987), 47–76 | MR | Zbl
[26] Okamoto K., “Studies on the Painlevé equations. III. Second and fourth Painlevé equations, P$_{\rm II}$ and P$_{\rm IV}$”, Math. Ann., 275 (1986), 221–255 | DOI | MR | Zbl
[27] Okamoto K., “Studies on the Painlevé equations. IV. Third Painlevé equation P$_{\rm III}$”, Funkcial. Ekvac., 30 (1987), 305–332 | MR | Zbl
[28] Quispel G. R. W., Roberts J. A. G., Thompson C. J., “Integrable mappings and soliton equations”, Phys. Lett. A, 126 (1988), 419–421 | DOI | MR | Zbl
[29] Quispel G. R. W., Roberts J. A. G., Thompson C. J., “Integrable mappings and soliton equations. II”, Phys. D, 34 (1989), 183–192 | DOI | MR | Zbl
[30] Ramani A., Grammaticos B., “Discrete Painlevé equations: coalescences, limits and degeneracies”, Phys. A, 228 (1996), 160–171, arXiv: solv-int/9510011 | DOI | MR | Zbl
[31] Sakai H., “Casorati determinant solutions for the $q$-difference sixth Painlevé equation”, Nonlinearity, 11 (1998), 823–833 | DOI | MR | Zbl
[32] Sakai H., “Rational surfaces associated with affine root systems and geometry of the Painlevé equations”, Comm. Math. Phys., 220 (2001), 165–229 | DOI | MR | Zbl
[33] Tsuda T., “Tau functions of $q$-Painlevé III and IV equations”, Lett. Math. Phys., 75 (2006), 39–47 | DOI | MR | Zbl
[34] Wadim Z., “Heine's basic transform and a permutation group for $q$-harmonic series”, Acta Arith., 111 (2004), 153–164 | DOI | MR | Zbl