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

SIGMA 8 (2012), 097, 27 pages      arXiv:1207.0041      https://doi.org/10.3842/SIGMA.2012.097

### Construction of a Lax Pair for the $\boldsymbol{E_6^{(1)}}$ $\boldsymbol{q}$-Painlevé System

Nicholas S. Witte a and Christopher M. Ormerod b
a) Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
b) Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia

Received September 05, 2012, in final form November 29, 2012; Published online December 11, 2012

Abstract
We construct a Lax pair for the $E^{(1)}_6$ $q$-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6$ $q$-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.

Key words: non-uniform lattices; divided-difference operators; orthogonal polynomials; semi-classical weights; isomonodromic deformations; Askey table.

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References

1. Birkhoff G.D., General theory of linear difference equations, Trans. Amer. Math. Soc. 12 (1911), 243-284.
2. Birkhoff G.D., The generalized Riemann problem for linear differential equations and the Allied problems for linear difference and $q$-difference equations, Trans. Amer. Math. Soc. 49 (1913), 521-568.
3. 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.
4. Jimbo M., Sakai H., A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154, chao-dyn/9507010.
5. 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), no. 24, 1439-1463, nlin.SI/0501051.
6. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Hypergeometric solutions to the $q$-Painlevé equations, Int. Math. Res. Not. 2004 (2004), no. 47, 2497-2521, arXiv:nlin.SI/0403036.
7. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
8. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998, http://aw.twi.tudelft.nl/~koekoek/askey/.
9. Magnus A.P., Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, in Orthogonal Polynomials and their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 261-278.
10. Magnus A.P., Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points, J. Comput. Appl. Math. 65 (1995), 253-265, math.CA/9502228.
11. Murata M., Lax forms of the $q$-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
12. Papageorgiou V.G., Nijhoff F.W., Grammaticos B., Ramani A., Isomonodromic deformation problems for discrete analogues of Painlevé equations, Phys. Lett. A 164 (1992), 57-64.
13. Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
14. Sakai H., A $q$-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273-297.
15. Sakai H., Lax form of the $q$-Painlevé equation associated with the $A^{(1)}_2$ surface, J. Phys. A: Math. Gen. 39 (2006), 12203-12210.
16. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
17. Witte N.S., Semi-classical orthogonal polynomial systems on non-uniform lattices, deformations of the Askey table and analogs of isomonodromy, arXiv:1204.2328.
18. Yamada Y., Lax formalism for $q$-Painlevé equations with affine Weyl group symmetry of type $E^{(1)}_n$, Int. Math. Res. Not. 2011 (2011), no. 17, 3823-3838, arXiv:1004.1687.