Geodesic
$q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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We provide an explicit expression for the first order $q$-difference system for the Jackson integral of symmetric Selberg type. The $q$-difference system gives a generalization of $q$-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the $q$-KZ equation. Our main result is an explicit expression for the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.
Keywords: $q$-difference equations, Selberg type integral, contiguous relations
Mots-clés : Gauss decomposition. 
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     author = {Masahiko Ito},
     title = {$q${-Difference} {Systems} for the {Jackson} {Integral} of {Symmetric} {Selberg} {Type}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a112/}
}
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Masahiko Ito. $q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a112/

[1] Albion S. P., Rains E. M., Warnaar S. O., AFLT-type Selberg integrals

[2] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[3] Aomoto K., “Jacobi polynomials associated with Selberg integrals”, SIAM J. Math. Anal., 18 (1987), 545–549 | DOI | MR | Zbl

[4] Aomoto K., “$q$-analogue of de Rham cohomology associated with Jackson integrals. I”, Proc. Japan Acad. Ser. A Math. Sci., 66 (1990), 161–164 | DOI | MR | Zbl

[5] Aomoto K., “Connection matrices and Riemann–Hilbert problem for $q$-difference equations”, Structure of Solutions of Differential Equations (Katata/Kyoto, 1995), World Sci. Publ., River Edge, NJ, 1996, 51–69 | MR | Zbl

[6] Aomoto K., “On elliptic product formulas for Jackson integrals associated with reduced root systems”, J. Algebraic Combin., 8 (1998), 115–126 | DOI | MR | Zbl

[7] Aomoto K., Kato Y., “A $q$-analogue of de Rham cohomology associated with Jackson integrals”, Special Functions, ICM-90 Satell. Conf. Proc. (Okayama, 1990), Tokyo, 1991, 30–62 | DOI | MR | Zbl

[8] Aomoto K., Kato Y., “Gauss decomposition of connection matrices and application to Yang–Baxter equation. I”, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 238–242 | DOI | MR | Zbl

[9] Aomoto K., Kato Y., “Gauss decomposition of connection matrices for symmetric $A$-type Jackson integrals”, Selecta Math. (N.S.), 1 (1995), 623–666 | DOI | MR | Zbl

[10] Aomoto K., Kato Y., “Derivation of $q$-difference equation from connection matrix for Selberg type Jackson integrals”, J. Differ. Equations Appl., 4 (1998), 247–278 | DOI | MR | Zbl

[11] Askey R., “Some basic hypergeometric extensions of integrals of Selberg and Andrews”, SIAM J. Math. Anal., 11 (1980), 938–951 | DOI | MR | Zbl

[12] Bosnjak G., Mangazeev V. V., “Construction of $R$-matrices for symmetric tensor representations related to $U_q(\widehat{sl_n})$”, J. Phys. A: Math. Theor., 49 (2016), 495204, 19 pp., arXiv: 1607.07968 | DOI | MR | Zbl

[13] Evans R. J., “Multidimensional $q$-beta integrals”, SIAM J. Math. Anal., 23 (1992), 758–765 | DOI | MR | Zbl

[14] Forrester P. J., Ito M., “Difference system for Selberg correlation integrals”, J. Phys. A: Math. Theor., 43 (2010), 175202, 19 pp., arXiv: 1011.1650 | DOI | MR | Zbl

[15] Forrester P. J., Warnaar S. O., “The importance of the Selberg integral”, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 489–534, arXiv: 0710.3981 | DOI | MR | Zbl

[16] Habsieger L., “Une $q$-intégrale de Selberg et Askey”, SIAM J. Math. Anal., 19 (1988), 1475–1489 | DOI | MR | Zbl

[17] Ishikawa M., Zeng J., Hankel hyperpfaffian calculations and Selberg integrals

[18] Ito M., Forrester P. J., “A bilateral extension of the $q$-Selberg integral”, Trans. Amer. Math. Soc., 369 (2017), 2843–2878, arXiv: 1309.0001 | DOI | MR | Zbl

[19] Ito M., Noumi M., “Connection formula for the Jackson integral of type $A_n$ and elliptic Lagrange interpolation”, SIGMA, 14 (2018), 077, 42 pp., arXiv: 1801.07041 | DOI | MR | Zbl

[20] Kadell K. W.J., “A proof of Askey's conjectured $q$-analogue of Selberg's integral and a conjecture of Morris”, SIAM J. Math. Anal., 19 (1988), 969–986 | DOI | MR | Zbl

[21] Kaneko J., “Selberg integrals and hypergeometric functions associated with Jack polynomials”, SIAM J. Math. Anal., 24 (1993), 1086–1110 | DOI | MR | Zbl

[22] Kaneko J., “$q$-Selberg integrals and Macdonald polynomials”, Ann. Sci. École Norm. Sup. (4), 29 (1996), 583–637 | DOI | MR | Zbl

[23] Kim J. S., Okada S., “A new $q$-Selberg integral, Schur functions, and Young books”, Ramanujan J., 42 (2017), 43–57, arXiv: 1412.7914 | DOI | MR | Zbl

[24] Kim J. S., Stanton D., “On $q$-integrals over order polytopes”, Adv. Math., 308 (2017), 1269–1317, arXiv: 1608.03342 | DOI | MR | Zbl

[25] Kim J. S., Yoo M., “Hook length property of $d$-complete posets via $q$-integrals”, J. Combin. Theory Ser. A, 162 (2019), 167–221, arXiv: 1708.09109 | DOI | MR | Zbl

[26] Kuniba A., Okado M., Yoneyama A., “Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q\big(A^{(1)}_{n-1}\big)$”, Lett. Math. Phys., 109 (2019), 2049–2067, arXiv: 1812.03767 | DOI | MR | Zbl

[27] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[28] Matsuo A., “Jackson integrals of Jordan–Pochhammer type and quantum Knizhnik–Zamolodchikov equations”, Comm. Math. Phys., 151 (1993), 263–273 | DOI | MR | Zbl

[29] Matsuo A., “Quantum algebra structure of certain Jackson integrals”, Comm. Math. Phys., 157 (1993), 479–498 | DOI | MR | Zbl

[30] Mimachi K., “Connection problem in holonomic $q$-difference system associated with a Jackson integral of Jordan–Pochhammer type”, Nagoya Math. J., 116 (1989), 149–161 | DOI | MR | Zbl

[31] Mimachi K., “Holonomic $q$-difference system of the first order associated with a Jackson integral of Selberg type”, Duke Math. J., 73 (1994), 453–468 | DOI | MR | Zbl

[32] Rimányi R., Tarasov V., Varchenko A., Zinn-Justin P., “Extended Joseph polynomials, quantized conformal blocks, and a $q$-Selberg type integral”, J. Geom. Phys., 62 (2012), 2188–2207, arXiv: 1110.2187 | DOI | MR | Zbl

[33] Varchenko A., “Quantized Knizhnik–Zamolodchikov equations, quantum Yang–Baxter equation, and difference equations for $q$-hypergeometric functions”, Comm. Math. Phys., 162 (1994), 499–528 | DOI | MR | Zbl

[34] Varchenko A., Special functions, KZ type equations, and representation theory, CBMS Regional Conference Series in Mathematics, 98, Amer. Math. Soc., Providence, RI, 2003 | DOI | MR | Zbl

[35] Warnaar S. O., “$q$-Selberg integrals and Macdonald polynomials”, Ramanujan J., 10 (2005), 237–268 | DOI | MR | Zbl