Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 084, 16 pages      arXiv:1008.2595     https://doi.org/10.3842/SIGMA.2010.084

Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2+A1)(1)

Nobutaka Nakazono
Graduate School of Mathematics, Kyushu University, 744 Motooka, Fukuoka, 819-0395, Japan

Received August 17, 2010, in final form October 08, 2010; Published online October 14, 2010

Abstract
We consider a q-Painlevé III equation and a q-Painlevé II equation arising from a birational representation of the affine Weyl group of type (A2+A1)(1). We study their hypergeometric solutions on the level of τ functions.

Key words: q-Painlevé system; hypergeometric function; affine Weyl group; τ function.

pdf (279 kb)   ps (178 kb)   tex (16 kb)

References

  1. Gasper G., Rahman, M. Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  2. Hamamoto T., Kajiwara K., Hypergeometric solutions to the q-Painlevé equation of type A4(1), J. Phys. A: Math. Theor. 40 (2007), 12509-12524, nlin.SI/0701001.
  3. Hamamoto T., Kajiwara K., Witte N.S., Hypergeometric solutions to the q-Painlevé equation of type (A1+A1')(1), Int. Math. Res. Not. 2006 (2006), Art. ID 84619, 26 pages, nlin.SI/0607065.
  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, nlin.SI/0512041.
  5. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Phys. D 2 (1981), 306-352.
  6. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  7. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981/82), 26-46.
  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, nlin.SI/0205019.
  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, solv-int/9903014.
  10. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., 10E9 solution to the elliptic Painlevé equation, J. Phys. A: Math. Gen. 36 (2003), L263-L272, nlin.SI/0303032.
  11. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Point configurations, Cremona transformations and the elliptic difference Painlevé equation, in Théories Asymptotiques et Équations de Painlevé, Semin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 169-198, nlin.SI/0411003.
  12. Kajiwara K., Nakazono N., Tsuda T., Projective reduction of the discrete Painlevé system of type (A2+A1)(1), Int. Math. Res. Not. 2010 (2010), Art. ID rnq089, 37 pages, arXiv:0910.4439.
  13. 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.
  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, solv-int/9709011.
  15. Kajiwara K., Ohta Y., Satsuma J., Casorati determinant solutions for the discrete Painlevé III equation, J. Math. Phys. 36 (1995), 4162-4174, solv-int/9412004.
  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, solv-int/9310002.
  17. Masuda T., Hypergeometric τ-functions of the q-Painlevé system of type E7(1), SIGMA 5 (2009), 035, 30 pages, arXiv:0903.4102.
  18. Masuda T., Hypergeometric τ-functions of the q-Painlevé system of type E8(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, Vol. 135, Cambridge University Press, Cambridge, 2000.
  20. Nakao S., Kajiwara K., Takahashi D., Multiplicative dPII 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, Vol. 223, American Mathematical Society, Providence, RI, 2004.
  22. Noumi M., Yamada Y., Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math. J. 153 (1999), 53-86, q-alg/9708018.
  23. Ohta Y., Nakamura A., Similarity KP equation and various different representations of its solutions, J. Phys. Soc. Japan 61 (1992), 4295-4313.
  24. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. 146 (1987), 337-381.
  25. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation PV, Japan. J. Math. (N.S.) 13 (1987), 47-76.
  26. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221-255.
  27. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305-332.
  28. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), 419-421.
  29. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations. II, Phys. D 34 (1989), 183-192.
  30. Ramani A., Grammaticos B., Discrete Painlevé equations: coalescences, limits and degeneracies, Phys. A 228 (1996), 160-171, solv-int/9510011.
  31. Sakai H., Casorati determinant solutions for the q-difference sixth Painlevé equation, Nonlinearity 11 (1998), 823-833.
  32. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  33. Tsuda T., Tau functions of q-Painlevé III and IV equations, Lett. Math. Phys. 75 (2006), 39-47.
  34. Wadim Z., Heine's basic transform and a permutation group for q-harmonic series, Acta Arith. 111 (2004), 153-164.

Previous article   Next article   Contents of Volume 6 (2010)