Geodesic
Hypergeometric $\tau$-Functions of the $q$-Painlevé System of Type $E_7^{(1)}$
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present the $\tau$-functions for the hypergeometric solutions to the $q$-Painlevé system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8W_7$. By using the $W(D_5)$ symmetry of the function ${}_8W_7$, we construct a set of twelve solutions and describe the action of $\widetilde W(D_6^{(1)})$ on the set.
Keywords: $q$-Painlevé system; $q$-hypergeometric function; Weyl group; $\tau$-function. 
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     author = {Tetsu Masuda},
     title = {Hypergeometric $\tau${-Functions} of the $q${-Painlev\'e} {System} of {Type} $E_7^{(1)}$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a34/}
}
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Tetsu Masuda. Hypergeometric $\tau$-Functions of the $q$-Painlevé System of Type $E_7^{(1)}$. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a34/

[1] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, 2004 | MR | Zbl

[2] Gupta D. P., Masson D. R., “Contiguous relations, continued fractions and orthogonality”, Trans. Amer. Math. Soc., 350 (1998), 769–808 ; math.CA/9511218 | DOI | MR | Zbl

[3] 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 ; nlin.SI/0701001 | DOI | MR | Zbl

[4] 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., ages ; nlin.SI/0607065 | DOI | MR

[5] Ismail M. E. H., Rahman M., “The associated Askey–Wilson polynomials”, Trans. Amer. Math. Soc., 328 (1991), 201–237 | DOI | MR | Zbl

[6] Kac V. G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990 | MR

[7] Kajiwara K., Kimura K., “On a $q$-difference Painlevé III equation. I. Derivation, symmetry and Riccati type solutions”, J. Nonlinear Math. Phys., 10 (2003), 86–102 ; nlin.SI/0205019 | DOI | MR | Zbl

[8] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “${}_{10}E_9$ solutions to the elliptic Painlevé equation”, J. Phys. A: Math. Gen., 36 (2003), L263–272 ; nlin.SI/0303032 | DOI | MR

[9] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “Hypergeometric solutions to the $q$-Painlevé equations”, Int. Math. Res. Not., 2004:47 (2004), 2497–2521 ; nlin.SI/0403036 | DOI | MR | Zbl

[10] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “Construction of hypergeometric solutions to the $q$-Painlevé equations”, Int. Math. Res. Not., 2005:24 (2005), 1439–1463 ; nlin.SI/0501051 | 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, Seminaires et Congres, Sémin. Congr., 14, eds. É. Delabaere et al., Soc. Math. France, Paris, 2006, 169–198 ; nlin.SI/0411003 | MR | Zbl

[12] Kajiwara K., Noumi M., Yamada Y., “A study on the fourth $q$-Painlevé equation”, J. Phys. A: Math. Gen., 34 (2001), 8563–8581 ; nlin.SI/0012063 | DOI | MR | Zbl

[13] Lievens S., Van der Jeugt J., “Invariance groups of three term transformations for basic hypergeometric series”, J. Comput. Appl. Math., 197 (2006), 1–14 | DOI | MR | Zbl

[14] Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E_8^{(1)}$, Preprint 2009-12, Kyushu University, 2009

[15] Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K. M., “Special function solutions of the discrete Painlevé equations”, Comp. Math. Appl., 42 (2001), 603–614 | DOI | MR | Zbl

[16] Sakai H., “Casorati determinant solutions for the $q$-difference sixth Painlevé equation”, Nonlinearity, 11 (1998), 823–833 | DOI | MR | Zbl

[17] 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

[18] Tsuda T., “Tropical Weyl group action via point configurations and $\tau$-functions of the $q$-Painlevé equations”, Lett. Math. Phys., 77 (2006), 21–30 | DOI | MR | Zbl