P. Baseilhac, Deformed Dolan???Grady relations in quantum integrable models, Nuclear Physics B, vol.709, issue.3, pp.491-521, 2005.
DOI : 10.1016/j.nuclphysb.2004.12.016

URL : https://hal.archives-ouvertes.fr/hal-00019441

P. Baseilhac, A family of tridiagonal pairs and related symmetric functions, Journal of Physics A: Mathematical and General, vol.39, issue.38, 2006.
DOI : 10.1088/0305-4470/39/38/005

URL : https://hal.archives-ouvertes.fr/hal-00082314

P. Baseilhac and S. Belliard, Generalized q-Onsager Algebras and Boundary Affine Toda Field Theories, Letters in Mathematical Physics, vol.89, issue.FS14, pp.213-228, 2010.
DOI : 10.1007/s11005-010-0412-6

. M. Ju and . Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Translated from the Russian by R, Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. Translations of Mathematical Monographs, vol.17, 1968.

P. Baseilhac and S. Belliard, A note on the Oq( sl2) algebra

P. Baseilhac and S. Belliard, The half-infinite XXZ chain in Onsager??s approach, Nuclear Physics B, vol.873, issue.3, pp.550-583, 2013.
DOI : 10.1016/j.nuclphysb.2013.05.003

E. Bannai and T. Ito, Algebraic Combinatorics I: Association schemes, 1984.

X. [. Baseilhac and . Martin, A bispectral q?hypergeometric basis for a class of quantum integrable models
URL : https://hal.archives-ouvertes.fr/hal-01168537

P. Baseilhac and T. T. Vu, -Onsager algebra, Journal of Mathematical Physics, vol.55, issue.8, p.81707, 2014.
DOI : 10.1063/1.4892518

URL : https://hal.archives-ouvertes.fr/hal-00556920

J. A. Charris, M. E. Ismail, and S. Monsalve, On sieved orthogonal polynomials. X. General blocks of recurrence relations, Pacific Journal of Mathematics, vol.163, issue.2, pp.163-165, 1994.
DOI : 10.2140/pjm.1994.163.237

V. Chari and A. Pressley, A guide to quantum groups. CUP, 1994.

T. Deguchi, K. Fabricius, and B. M. Mccoy, The sl2 loop algebra symmetry of the six-vertex model at roots of unity, Journal of Statistical Physics, vol.102, issue.3/4, pp.701-736, 2001.
DOI : 10.1023/A:1004894701900

L. Dolan and M. Grady, Conserved charges from self-duality, Physical Review D, vol.25, issue.6, p.1587, 1982.
DOI : 10.1103/PhysRevD.25.1587

A. J. Durán and W. Van-assche, Orthogonal matrix polynomials and higher-order recurrence relations, Linear Algebra and its Applications, vol.219, pp.261-280, 1995.
DOI : 10.1016/0024-3795(93)00218-O

J. S. Geronimo and P. Iliev, Bispectrality of Multivariable Racah???Wilson Polynomials, Constructive Approximation, vol.141, issue.2, pp.417-457, 2010.
DOI : 10.1007/s00365-009-9045-3

W. Groenevelt, M. E. Ismail, and E. Koelink, Spectral decomposition and matrix-valued orthogonal polynomials, Advances in Mathematics, vol.244, pp.91-1051206, 2013.
DOI : 10.1016/j.aim.2013.04.025

Y. I. Granovskii, I. M. Lutzenko, and A. S. Zhedanov, Mutual integrability, quadratic algebras, and dynamical symmetry, Annals of Physics, vol.217, issue.1, pp.1-20, 1992.
DOI : 10.1016/0003-4916(92)90336-K

G. Gasper and M. Rahman, Some Systems of Multivariable Orthogonal Askey-Wilson Polynomials, Dev. Math, vol.13, pp.209-219, 2005.
DOI : 10.1007/0-387-24233-3_10

P. Iliev, Bispectral commuting difference operators for multivariable Askey-Wilson polynomials, Transactions of the American Mathematical Society, vol.363, issue.03, pp.1577-1598, 2011.
DOI : 10.1090/S0002-9947-2010-05183-9

T. Ito, K. Nomura, and P. Terwilliger, A classification of sharp tridiagonal pairs, Linear Algebra and its Applications, vol.435, issue.8, pp.1857-1884, 2011.
DOI : 10.1016/j.laa.2011.03.032

T. Ito and P. Terwilliger, Mock tridiagonal systems, Linear Algebra and its Applications, vol.435, issue.8, 1997.
DOI : 10.1016/j.laa.2011.03.025

T. Ito, P. Terwilliger, ]. T. Ito, K. Tanabe, and P. Terwilliger, The augmented tridiagonal algebra Some algebra related to P -and Q-polynomial association schemes, Codes and association schemes, Kyushu J. Math. DIMACS Ser. Discrete Math. Theoret. Comput. Sci, vol.64, issue.56, pp.81-144, 1999.

R. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q?analogue, arXiv:math.CA, 96022141.

M. Krein, Infinite J?matrices and a matrix-moment problem, pp.125-128, 1949.

S. Kolb, Quantum symmetric Kac???Moody pairs, Advances in Mathematics, vol.267, pp.395-469, 2014.
DOI : 10.1016/j.aim.2014.08.010

D. Leonard, Orthogonal Polynomials, Duality and Association Schemes, SIAM Journal on Mathematical Analysis, vol.13, issue.4, pp.656-663, 1982.
DOI : 10.1137/0513044

G. Luztig, Introduction to Quantum Groups, Birkhauser, 1993.
DOI : 10.1007/978-0-8176-4717-9

K. Nomura and P. Terwilliger, Tridiagonal pairs of q-Racah type and the µ-conjecture, Linear Algebra Appl, vol.432, pp.320-3209, 2010.

L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Physical Review, vol.65, issue.3-4, pp.117-149, 1944.
DOI : 10.1103/PhysRev.65.117

P. Terwilliger, The subconstituent algebra of an association scheme. III, Journal of Algebraic Combinatorics, vol.2, issue.2, pp.177-210, 1993.
DOI : 10.1023/A:1022415825656

P. Terwilliger, N. Kirillov, A. Tsuchiya, and H. Umemura, Two relations that generalize the q?Serre relations and the Dolan-Grady relations, Proceedings of the Nagoya 1999 International workshop on physics and combinatorics, pp.377-398

P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra and its Applications, vol.330, issue.1-3, pp.149-203, 2001.
DOI : 10.1016/S0024-3795(01)00242-7

P. Terwilliger, Leonard pairs and the q-Racah polynomials, Linear Algebra and its Applications, vol.387, pp.235-276, 2004.
DOI : 10.1016/j.laa.2004.02.014

P. Terwilliger, Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array, Designs, Codes and Cryptography, pp.307-332, 2005.

M. V. Tratnik and M. Wilson-polynomials, Multivariable Meixner, Krawtchouk, and Meixner???Pollaczek polynomials, Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials, pp.2740-2749, 1989.
DOI : 10.1063/1.528507

P. Terwilliger and R. Vidunas, LEONARD PAIRS AND THE ASKEY???WILSON RELATIONS, Journal of Algebra and Its Applications, vol.03, issue.04, 2004.
DOI : 10.1142/S0219498804000940

S. Tsujimoto, L. Vinet, and A. Zhedanov, Dual $-1$ Hahn polynomials: ???Classical??? polynomials beyond the Leonard duality, Proc. Amer, pp.959-970, 2013.
DOI : 10.1090/S0002-9939-2012-11469-8

D. Uglov and L. Ivanov, sl(N) Onsager's algebra and integrability, Journal of Statistical Physics, vol.136, issue.F14, pp.hep-th, 1996.
DOI : 10.1007/BF02189226

T. Vu, The higher order q?Dolan-Grady relations and quantum integrable systems, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01245741

A. S. Zhedanov, ?Hidden symmetry? of Askey-Wilson polynomials, Theoretical and Mathematical Physics, vol.22, issue.2, pp.190-204, 1991.
DOI : 10.1007/BF01015906

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