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

SIGMA 14 (2018), 011, 32 pages      arXiv:1609.02525      https://doi.org/10.3842/SIGMA.2018.011
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Series Solutions of the Non-Stationary Heun Equation

Farrokh Atai ab and Edwin Langmann a
a) Department of Physics, KTH Royal Institute of Technology, SE-10691 Stockholm, Sweden
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received October 10, 2017, in final form February 08, 2018; Published online February 16, 2018

Abstract
We consider the non-stationary Heun equation, also known as quantum Painlevé VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the problem to solve this equation into a differential-difference equation which, as we show, can be solved by efficient recursive algorithms. We thus obtain series representations of solutions which provide elliptic generalizations of the Jacobi polynomials. These series reproduce, in a limiting case, a perturbative solution of the Heun equation due to Takemura, but our method is different in that we expand in non-conventional basis functions that allow us to obtain explicit formulas to all orders; in particular, for special parameter values, our series reduce to a single term.

Key words: Heun equation; Lamé equation; Kernel functions; quantum Painlevé VI; perturbation theory.

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References

1. Bazhanov V.V., Mangazeev V.V., Eight-vertex model and non-stationary Lamé equation, J. Phys. A: Math. Gen. 38 (2005), L145-L153, hep-th/0411094.
2. Chalykh O., Feigin M., Veselov A., New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys. 39 (1998), 695-703.
3. Erdélyi A., Integral equations for Heun functions, Quart. J. Math., Oxford Ser. 13 (1942), 107-112.
4. Etingof P.I., Frenkel I.B., Kirillov Jr. A.A., Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), 59-90, hep-th/9407047.
5. Etingof P.I., Kirillov Jr. A.A., Representations of affine Lie algebras, parabolic differential equations, and Lamé functions, Duke Math. J. 74 (1994), 585-614, hep-th/9310083.
6. Falceto F., Gawędzki K., Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe ansatz for the elliptic Hitchin systems, Comm. Math. Phys. 183 (1997), 267-290, hep-th/9604094.
7. Fateev V.A., Litvinov A.V., Neveu A., Onofri E., A differential equation for a four-point correlation function in Liouville field theory and elliptic four-point conformal blocks, J. Phys. A: Math. Theor. 42 (2009), 304011, 29 pages, arXiv:0902.1331.
8. Felder G., Varchenko A., Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Int. Math. Res. Not. 1995 (1995), 221-233, hep-th/9502165.
9. Felder G., Varchenko A., Special functions, conformal blocks, Bethe ansatz and ${\rm SL}(3,{\mathbb Z})$, Phil. Trans. R. Soc. Lond. A 359 (2001), 1365-1373, math.QA/0101136.
10. Hallnäs M., Langmann E., A unified construction of generalized classical polynomials associated with operators of Calogero-Sutherland type, Constr. Approx. 31 (2010), 309-342, math-ph/0703090.
11. Hallnäs M., Ruijsenaars S., A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic Calogero-Moser Hamiltonians, Int. Math. Res. Not. 2015 (2015), 10278-10313, arXiv:1305.4759.
12. Inozemtsev V.I., Lax representation with spectral parameter on a torus for integrable particle systems, Lett. Math. Phys. 17 (1989), 11-17.
13. Kazakov A.Y., Slavyanov S.Yu., Integral relations for Heun-class special functions, Theoret. and Math. Phys. 107 (1996), 733-739.
14. Kolb S., Radial part calculations for $\widehat{\mathfrak{sl}}_2$ and the Heun-KZB heat equation, Int. Math. Res. Not. 2015 (2015), 12941-12990, arXiv:1310.0782.
15. Komori Y., Noumi M., Shiraishi J., Kernel functions for difference operators of Ruijsenaars type and their applications, SIGMA 5 (2009), 054, 40 pages, arXiv:0812.0279.
16. Komori Y., Takemura K., The perturbation of the quantum Calogero-Moser-Sutherland system and related results, Comm. Math. Phys. 227 (2002), 93-118, math.QA/0009244.
17. Koroteev P., Sciarappa A., On elliptic algebras and large-$n$ supersymmetric gauge theories, J. Math. Phys. 57 (2016), 112302, 32 pages, arXiv:1601.08238.
18. Lambe C.G., Ward D.R., Some differential equations and associated integral equations, Quart. J. Math. os-5 (1934), 81-97.
19. Langmann E., Anyons and the elliptic Calogero-Sutherland model, Lett. Math. Phys. 54 (2000), 279-289, math-ph/0007036.
20. Langmann E., Algorithms to solve the (quantum) Sutherland model, J. Math. Phys. 42 (2001), 4148-4157, math-ph/0104039.
21. Langmann E., Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model, J. Math. Phys. 47 (2006), 022101, 18 pages, math-ph/0406061.
22. Langmann E., Singular eigenfunctions of Calogero-Sutherland type systems and how to transform them into regular ones, SIGMA 3 (2007), 031, 18 pages, math-ph/0702089.
23. Langmann E., Source identity and kernel functions for elliptic Calogero-Sutherland type systems, Lett. Math. Phys. 94 (2010), 63-75, arXiv:1003.0857.
24. Langmann E., Explicit solution of the (quantum) elliptic Calogero-Sutherland model, Ann. Henri Poincaré 15 (2014), 755-791, math-ph/0407050.
25. Langmann E., Takemura K., Source identity and kernel functions for Inozemtsev-type systems, J. Math. Phys. 53 (2012), 082105, 19, arXiv:1202.3544.
26. Maier R.S., The 192 solutions of the Heun equation, Math. Comp. 76 (2007), 811-843, math.CA/0408317.
27. Nagoya H., Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations, J. Math. Phys. 52 (2011), 083509, 16 pages, arXiv:1109.1645.
28. Nawata S., Givental $J$-functions, quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys. 19 (2015), 1277-1338, arXiv:1408.4132.
29. Nekrasov N.A., Shatashvili S.L., Quantization of integrable systems and four dimensional gauge theories, in XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, 265-289, arXiv:0908.4052.
30. NIST digital library of mathematical functions, Release date 2010-05-07, available at http://dlmf.nist.gov/.
31. Novikov D.P., Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation, Theoret. and Math. Phys. 146 (2006), 295-303.
32. Ronveaux A., Arscott F. (Editors), Heun's differential equations, Oxford Science Publications, Oxford University Press, Oxford, 1995.
33. Rosengren H., Special polynomials related to the supersymmetric eight-vertex model. III. Painlevé VI equation, arXiv:1405.5318.
34. Rosengren H., Special polynomials related to the supersymmetric eight-vertex model: a summary, Comm. Math. Phys. 340 (2015), 1143-1170, arXiv:1503.02833.
35. Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. III. The Heun case, SIGMA 5 (2009), 049, 21 pages, arXiv:0904.3250.
36. Sergeev A.N., Calogero operator and Lie superalgebras, Theoret. and Math. Phys. 131 (2002), 747-764.
37. Slavyanov S.Yu., Lay W., Special functions. A unified theory based on singularities, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
38. Smirnov A.O., Elliptic solitons and Heun's equation, in The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, Vol. 32, Amer. Math. Soc., Providence, RI, 2002, 287-305, math.CA/0109149.
39. Takemura K., The Heun equation and the Calogero-Moser-Sutherland system. II. Perturbation and algebraic solution, Electron. J. Differential Equations 2004 (2004), 15, 30 pages, math.CA/0112179.
40. Takemura K., The Heun equation and the Calogero-Moser-Sutherland system. IV. The Hermite-Krichever ansatz, Comm. Math. Phys. 258 (2005), 367-403, math.CA/0406141.
41. Whittaker E.T., Watson G.N., A course of modern analysis. 4th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.