The generating function of bivariate Chebyshev polynomials associated
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 207-220 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct the generating function of the second-kind bivariate Chebyshev polynomials associated with the simple Lie algebra $G_2$ using a previously proposed method.
Keywords: bivariate Chebyshev polynomial, generating function.
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E. V. Damaskinsky; M. A. Sokolov. The generating function of bivariate Chebyshev polynomials associated. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 207-220. http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a1/

[1] E. V. Damaskinskii, P. P. Kulish, M. A. Sokolov, O vychislenii proizvodyaschikh funktsii obobschennykh polinomov Chebysheva neskolkikh peremennykh, preprint 13/2014, POMI, SPb., 2014

[2] E. V. Damaskinsky, P. P. Kulish, M. A. Sokolov, “On calculation of generating functions of Chebyshev polynomials in several variables”, J. Math. Phys., 56:6 (2015), 063507, 11 pp., arXiv: 1502.08000 | DOI | MR

[3] M. A. Sokolov, “Proizvodyaschie funktsii polinomov Chebysheva trekh peremennykh”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 23, Zap. nauchn. sem. POMI, 433, POMI, SPb., 2015, 246–259, arXiv: 1502.08027 | DOI | MR

[4] P. P. Kulish, “Integrable spin chains and representation theory”, Symmetries and Groups in Contemporary Physics, Proceedings of the XXIX International Colloquium on Group–Theoretical Methods in Physics (Tianjin, China, 20–26 August, 2012), Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 11, eds. C. Bai, J.-P. Gazeau, M.-L. Ge, World Sci., Singapore, 2013, 487–492 | DOI | MR

[5] T. H. Koornwinder, “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I”, Indag. Math., 77:1 (1974), 48–58 | DOI | MR

[6] P. K. Suetin, Ortogonalnye mnogochleny po dvum peremennym, Nauka, M., 1988 | MR | Zbl

[7] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979 | MR | Zbl

[8] T. J. Rivlin, The Chebyshev Polynomials, John Wiley and Sons, New York, 1974 | MR

[9] B. N. Ryland, H. Z. Munthe-Kaas, “On multivariate Chebyshev polynomials and spectral approximations on triangles”, Spectral and High Order Methods for Partial Differential Equations (Trondheim, Norway, June 22–26, 2009), v. 76, Lecture Notes in Computer Science and Engineering, eds. J. S. Hesthaven, E. M. Rønquist, Springer, Berlin, 2011, 19–41 | DOI | MR

[10] B. Shapiro, M. Shapiro, “On eigenvalues of rectangular matrices”, Tr. MIAN, 267 (2009), 258–265 | DOI | MR | Zbl

[11] P. Alexandersson, B. Shapiro, “Around a multivariate Schmidt–Spitzer theorem”, Linear Algebra Appl., 446 (2014), 356–368 | DOI | MR | Zbl

[12] P. P. Kulish, V. D. Lyakhovskii, O. V. Postnova, “Funktsiya kratnostei dlya tenzornykh stepenei modulei algebry $A_n$”, TMF, 171:2 (2012), 283–293 | DOI

[13] P. P. Kulish, V. D. Lyakhovsky, O. V. Postnova, “Tensor power decomposition. $B_n$-case”, J. Phys.: Conf. Ser., 343:1 (2012), 012095, 7 pp. | DOI

[14] V. D. Lyakhovskii, “Mnogochleny Chebysheva ot mnogikh peremennykh v terminakh singulyarnykh elementov”, TMF, 175:3 (2013), 419–428 | DOI | DOI | MR | Zbl

[15] G. von Gehlen, S. Roan, “The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials”, Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, v. 35, NATO Science Series, eds. S. Pakuliak, G. Von Gehlen, Springer, Berlin, 2001, 155–172 | MR

[16] G. von Gehlen, “Onsager's algebra and partially orthogonal polynomials”, Internat. J. Modern Phys. B, 16:14n15 (2002), 2129–2136 | DOI | MR

[17] V. V. Borzov, E. V. Damaskinskii, “Ostsillyator Chebysheva–Koornvindera”, TMF, 175:3 (2013), 765–772 | DOI | DOI | MR | Zbl

[18] V. V. Borzov, E. V. Damaskinsky, “The algebra of two dimensional generalized Chebyshev–Koornwinder oscillator”, J. Math. Phys., 55:10 (2014), 103505, 23 pp. | DOI | MR

[19] G. J. Heckman, “Root systems and hypergeometric functions. II”, Compositio Math., 64:3 (1987), 353–373 | MR | Zbl

[20] M. E. Hoffman, W. D. Withers, “Generalized Chebyshev polynomials associated with affine Weyl groups”, Trans. Amer. Math. Soc., 308:1 (1988), 91–104 | DOI | MR | Zbl

[21] R. J. Beerends, “Chebyshev polynomials in several variables and the radial part Laplace–Beltrami operator”, Trans. Amer. Math. Soc., 328:2 (1991), 779–814 | DOI | MR

[22] A. Klimyk, J. Patera, “Orbit functions”, SIGMA, 2 (2006), 006, 60 pp. | DOI | MR | Zbl

[23] V. D. Lyakhovsky, Ph. V. Uvarov, “Multivariate Chebyshev polynomials”, J. Phys. A: Math. Theor., 46:12 (2013), 125201, 22 pp. | DOI | MR

[24] K. B. Dunn, R. Lidl, “Generalizations of the classical Chebyshev polynomials to polynomials in two variables”, Czechoslovak Math. J., 32:4 (1982), 516–528 | MR | Zbl

[25] P. P. Kulish, “Models solvable by Bethe ansatz”, J. Gen. Lie Theory Appl., 2:3 (2008), 190–200 | DOI | MR | Zbl

[26] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1979), 194–220 | DOI | MR

[27] P. P. Kulish, E. K. Sklyanin, “Quantum spectral transform methods. Recent developments”, Integrable Quantum Field Theories (Tvärminne, Finland, 23–27 March, 1981), Lecture Notes in Physics, 151, eds. J. Hietarinta, C. Montonen, Springer, Berlin, 61–119 | DOI | MR

[28] P. P. Kulish, N. Yu. Reshetikhin, “Kvantovaya lineinaya zadacha dlya uravneniya sinus-Gordon i vysshie predstavleniya”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 2, v. 101, Zap. nauchn. sem. LOMI, Nauka, L., 1981, 101–110 | DOI | MR

[29] V. G. Drinfeld, “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, CA, August 3–11, 1986), ed. A. M. Gleason, AMS, Providence, RI, 1987, 798–820 | MR | Zbl

[30] N. Yu. Reshetikhin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovanie grupp Li i algebr Li”, Algebra i analiz, 1:1 (1989), 178–206 | MR | Zbl

[31] L. A. Takhtadzhyan, L. D. Faddeev, “Spektr i rasseyanie vozbuzhdenii v odnomernom izotropnom magnetike Geizenberga”, Differentsialnaya geometriya, gruppy Li i mekhanika. IV, Zap. nauchn. sem. LOMI, 109, Nauka, L., 1981, 134–178 | DOI | MR | Zbl

[32] H. N. V. Temperley, E. H. Lieb, “Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem”, Proc. Roy. Soc. London Ser. A, 322:1549 (1971), 251–280 | DOI | MR | Zbl

[33] P. P. Martin, Potts Models and Related Problems in Statistical Mechanics, Series on Advances in Statistical Mechanics, 5, World Sci., Singapore, 1991 | MR

[34] R. Brauer, “On algebras which are connected with the semisimple continuous groups”, Ann. Math. (2), 38:4 (1937), 857–872 | DOI | MR | Zbl

[35] H. Wenzl, “On the structure of Brauer's centralizer algebras”, Ann. Math., 128:1 (1988), 173–193 | DOI | MR | Zbl

[36] J. S. Birman, H. Wenzl, “Braids, link polynomials and a new algebra”, Trans. Amer. Math. Soc., 313:1 (1989), 249–273 | DOI | MR | Zbl

[37] J. Murakami, “The Kauffman polynomial of links and representation theory”, Osaka J. Math., 24:4 (1987), 745–758 | MR | Zbl

[38] P. P. Kulish, “On spin systems related to the Temperley–Lieb algebra”, J. Phys. A: Math. Gen., 36:38 (2003), L489–L493 | DOI | MR | Zbl