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Kernel Function, $q$-Integral Transformation and $q$-Heun Equations
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the perspective of the $q$-integral transformation.
Keywords: kernel function, Jackson integral, Ruijsenaars system.
Mots-clés : Heun equation, $q$-Heun equation 
@article{SIGMA_2024_20_a82,
     author = {Kouichi Takemura},
     title = {Kernel {Function,} $q${-Integral} {Transformation} and $q${-Heun} {Equations}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a82/}
}
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Kouichi Takemura. Kernel Function, $q$-Integral Transformation and $q$-Heun Equations. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a82/

[1] Arai Y., Takemura K., “On $q$-middle convolution and $q$-hypergeometric equations”, SIGMA, 19 (2023), 037, 40 pp., arXiv: 2209.02227 | DOI | MR | Zbl

[2] Atai F., Noumi M., “Eigenfunctions of the van Diejen model generated by gauge and integral transformations”, Adv. Math., 412 (2023), 108816, 60 pp., arXiv: 2203.00498 | DOI | MR | Zbl

[3] Fujii T., Nobukawa T., Hypergeometric solutions for variants of the $q$-hypergeometric equation, arXiv: 2207.12777

[4] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[5] Hahn W., “On linear geometric difference equations with accessory parameters”, Funkcial. Ekvac., 14 (1971), 73–78 | MR | Zbl

[6] Hatano N., Matsunawa R., Sato T., Takemura K., “Variants of $q$-hypergeometric equation”, Funkcial. Ekvac., 65 (2022), 159–190, arXiv: 1910.12560 | DOI | MR | Zbl

[7] Kazakov A.Ya., “Integral symmetries, integral invariants, and monodromy matrices for ordinary differential equations”, Theoret. and Math. Phys., 116 (1998), 991–1000 | DOI | MR | Zbl

[8] Kazakov A.Ya., Slavyanov S.Yu., “Integral relations for special functions of the Heun class”, Theoret. and Math. Phys., 107 (1996), 733–739 | DOI | MR | Zbl

[9] Kojima K., Sato T., Takemura K., “Polynomial solutions of $q$-Heun equation and ultradiscrete limit”, J. Difference Equ. Appl., 25 (2019), 647–664, arXiv: 1809.01428 | DOI | MR | Zbl

[10] Komori Y., Noumi M., Shiraishi J., “Kernel functions for difference operators of Ruijsenaars type and their applications”, SIGMA, 5 (2009), 054, 40 pp., arXiv: 0812.0279 | DOI | MR | Zbl

[11] Langmann E., “Source identity and kernel functions for elliptic Calogero–Sutherland type systems”, Lett. Math. Phys., 94 (2010), 63–75, arXiv: 1003.0857 | DOI | MR | Zbl

[12] Langmann E., Takemura K., “Source identity and kernel functions for Inozemtsev-type systems”, J. Math. Phys., 53 (2012), 082105, 19 pp., arXiv: 1202.3544 | DOI | MR | Zbl

[13] Ruijsenaars S.N.M., “Integrable $BC_N$ analytic difference operators: hidden parameter symmetries and eigenfunctions”, New Trends in Integrability and Partial Solvability, NATO Sci. Ser. II Math. Phys. Chem., 132, Kluwer, Dordrecht, 2004, 217–261 | DOI | MR | Zbl

[14] Ruijsenaars S.N.M., “Zero-eigenvalue eigenfunctions for differences of elliptic relativistic Calogero–Moser Hamiltonians”, Theoret. and Math. Phys., 146 (2006), 25–33 | DOI | MR | Zbl

[15] Ruijsenaars S.N.M., “Hilbert–Schmidt operators vs. integrable systems of elliptic Calogero–Moser type. I The eigenfunction identities”, Comm. Math. Phys., 286 (2009), 629–657 | DOI | MR | Zbl

[16] Sakai H., Yamaguchi M., “Spectral types of linear $q$-difference equations and $q$-analog of middle convolution”, Int. Math. Res. Not., 2017 (2017), 1975–2013, arXiv: 1410.3674 | DOI | MR | Zbl

[17] Sasaki S., Takagi S., Takemura K., “$q$-middle convolution and $q$-Painlevé equation”, SIGMA, 18 (2022), 056, 21 pp., arXiv: 2201.03960 | DOI | MR | Zbl

[18] Sasaki S., Takagi S., Takemura K., “$q$-Heun equation and initial-value space of $q$-Painlevé equation”, Recent Trends in Formal and Analytic Solutions of Diff. Equations, Contemp. Math., 782, American Mathematical Society, Providence, RI, 2023, 119–142, arXiv: 2110.13860 | DOI | MR | Zbl

[19] Takemura K., “Integral representation of solutions to Fuchsian system and Heun's equation”, J. Math. Anal. Appl., 342 (2008), 52–69, arXiv: 0705.3358 | DOI | MR | Zbl

[20] Takemura K., “Middle convolution and Heun's equation”, SIGMA, 5 (2009), 040, 22 pp., arXiv: 0810.3112 | DOI | MR | Zbl

[21] Takemura K., “Heun's differential equation”, Selected Papers on Analysis and Differential Equations, Amer. Math. Soc. Transl., 230, 2nd ed., American Mathematical Society, Providence, RI, 2010, 45–68 | DOI | MR

[22] Takemura K., “Degenerations of Ruijsenaars–van Diejen operator and $q$-Painlevé equations”, J. Integrable Syst., 2 (2017), xyx008, 27 pp., arXiv: 1608.07265 | DOI | MR | Zbl

[23] Takemura K., “Integral transformation of Heun's equation and some applications”, J. Math. Soc. Japan, 69 (2017), 849–891, arXiv: 1008.4007 | DOI | MR | Zbl

[24] Takemura K., “On $q$-deformations of the Heun equation”, SIGMA, 14 (2018), 061, 16 pp., arXiv: 1712.09564 | DOI | MR | Zbl

[25] van Diejen J.F., “Integrability of difference Calogero–Moser systems”, J. Math. Phys., 35 (1994), 2983–3004 | DOI | MR | Zbl