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

SIGMA 10 (2014), 090, 23 pages      arXiv:1301.2401      https://doi.org/10.3842/SIGMA.2014.090

### Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlevé IV Equation

Nobutaka Nakazono
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia

Received June 06, 2013, in final form August 14, 2014; Published online August 22, 2014

Abstract
We consider a $q$-Painlevé IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ${}_2\varphi_1$ basic hypergeometric series and the other is given by ${}_2\psi_2$ bilateral basic hypergeometric series.

Key words: $q$-Painlevé equation; basic hypergeometric function; affine Weyl group; $\tau$-function; projective reduction; orthogonal polynomial.

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References

1. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
2. Hamamoto T., Kajiwara K., Hypergeometric solutions to the $q$-Painlevé equation of type $A_4^{(1)}$, J. Phys. A: Math. Gen. 40 (2007), 12509-12524, nlin.SI/0701001.
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), 84619, 26 pages, nlin.SI/0607065.
4. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
5. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
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), 26-46.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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, nlin.SI/0303032.
13. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2004 (2004), 2497-2521, nlin.SI/0403036.
14. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Construction of hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2005 (2005), 1441-1463, nlin.SI/0501051.
15. 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, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 169-198, nlin.SI/0411003.
16. Kajiwara K., Nakazono N., Hypergeometric solutions to the symmetric $q$-Painlevé equations, Int. Math. Res. Not., to appear, arXiv:1304.0858.
17. Kajiwara K., Nakazono N., Tsuda T., Projective reduction of the discrete Painlevé system of type $(A_2+A_1)^{(1)}$, Int. Math. Res. Not. (2011), 930-966, arXiv:0910.4439.
18. 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.
19. 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.
20. 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.
21. 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.
22. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
23. Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E_7^{(1)}$, SIGMA 5 (2009), 035, 30 pages, arXiv:0903.4102.
24. Masuda T., Hypergeometric $\tau$-functions of the $q$-Painlevé system of type $E^{(1)}_8$, Ramanujan J. 24 (2011), 1-31.
25. 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.
26. Nakazono N., Hypergeometric $\tau$ functions of the $q$-Painlevé systems of type $(A_2+A_1)^{(1)}$, SIGMA 6 (2010), 084, 16 pages, arXiv:1008.2595.
27. Noumi M., Painlevé equations through symmetry, Translations of Mathematical Monographs, Vol. 223, Amer. Math. Soc., Providence, RI, 2004.
28. Ohta Y., Nakamura A., Similarity KP equation and various different representations of its solutions, J. Phys. Soc. Japan 61 (1992), 4295-4313.
29. 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.
30. Okamoto K., Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{{\rm VI}}$, Ann. Mat. Pura Appl. 146 (1987), 337-381.
31. Okamoto K., Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\rm V}$, Japan. J. Math. (N.S.) 13 (1987), 47-76.
32. Okamoto K., Studies on the Painlevé equations. IV. Third Painlevé equation $P_{{\rm III}}$, Funkcial. Ekvac. 30 (1987), 305-332.
33. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), 419-421.
34. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations. II, Phys. D 34 (1989), 183-192.
35. Ramani A., Grammaticos B., Discrete Painlevé equations: coalescences, limits and degeneracies, Phys. A 228 (1996), 160-171, solv-int/9510011.
36. Sakai H., Casorati determinant solutions for the $q$-difference sixth Painlevé equation, Nonlinearity 11 (1998), 823-833.
37. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
38. Tamizhmani K.M., Grammaticos B., Carstea A.S., Ramani A., The $q$-discrete Painlevé IV equations and their properties, Regul. Chaotic Dyn. 9 (2004), 13-20.
39. Tsuda T., Tau functions of $q$-Painlevé III and IV equations, Lett. Math. Phys. 75 (2006), 39-47.
40. Uvarov V.B., The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Comput. Math. Math. Phys. 9 (1969), no. 6, 25-36.
41. Zhedanov A., Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), 67-86.