Abstract
We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type \(E_{8}^{(1)}\) in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct 56 solutions and describe the action of \(W(E_{7}^{(1)})\) on the solutions.
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References
Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 96. Cambridge University Press, Cambridge (2004)
Gupta, D.P., Masson, D.R.: Contiguous relations, continued fractions and orthogonality. Trans. Am. Math. Soc. 350, 769–808 (1998)
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), Art. ID 84619, 26 pp.
Kajiwara, K., Kimura, K.: On a q-difference Painlevé III equation, I. Derivation, symmetry and Riccati type solutions. J. Nonlinear Math. Phys. 10, 86–102 (2003)
Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: 10 E 9 solutions to the elliptic Painlevé equation. J. Phys. A 36, L263–272 (2003)
Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: Hypergeometric solutions to the q-Painlevé equations. Int. Math. Res. Not. 47, 2497–2521 (2004)
Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: Construction of hypergeometric solutions to the q-Painlevé equations. Int. Math. Res. Not. 24, 1439–1463 (2005)
Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: Point configurations, Cremona transformations and the elliptic difference Painlevé equation. Sémin. Congr. 14, 175–204 (2006)
Kajiwara, K., Noumi, M., Yamada, Y.: A study on the fourth q-Painlevé equation. J. Phys. A 34, 8563–8581 (2001)
Kurokawa, N., Koyama, S.: Multiple sine functions. Forum Math. 15, 839–876 (2003)
Lievens, S., Van der Jeugt, J.: Symmetry groups of Bailey’s transformations for 10 φ 9-series. J. Comput. Appl. Math. 206, 498–519 (2007)
Murata, M., Sakai, H., Yoneda, J.: Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type \(E^{(1)}_{8}\). J. Math. Phys. 44, 1396–1414 (2003)
Ohta, Y., Ramani, A., Grammaticos, B.: An affine Weyl group approach to the eight-parameter discrete Painlevé equation. J. Phys. A 34, 10523–10532 (2001)
Sakai, H.: Casorati determinant solutions for the q-difference sixth Painlevé equation. Nonlinearity 11, 823–833 (1998)
Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys. 220, 165–229 (2001)
Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24, 167–199 (1977)
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Masuda, T. Hypergeometric τ-functions of the q-Painlevé system of type \(E_{8}^{(1)}\) . Ramanujan J 24, 1–31 (2011). https://doi.org/10.1007/s11139-010-9262-1
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DOI: https://doi.org/10.1007/s11139-010-9262-1