For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In the first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [Dev. Math. 13, 209 (2005)] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [Trans. Am. Math. Soc. 363, 1577 (2011)]. Raising and lowering operators extending those of Sahi [SIGMA 3, 002 (2007)] are also constructed. In the second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In the third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie’s relations. In the fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q ≠ 1).

1.
G.
Gasper
and
M.
Rahman
, “
Some systems of multivariable orthogonal Askey-Wilson polynomials
,”
Dev. Math.
13
,
209
219
(
2005
); e-print arXiv:math/0410249.
2.
P.
Iliev
, “
Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
,”
Trans. Am. Math. Soc.
363
,
1577
1598
(
2011
); e-print arXiv:0801.4939.
3.
S.
Sahi
, “
Raising and lowering operators for Askey–Wilson polynomials
,”
SIGMA
3
,
002
(
2007
); e-print arXiv:math.QA/0701134.
4.
N. I.
Stoilova
and
J.
Van der Jeugt
, “
An exactly solvable spin chain related to Hahn polynomials
,”
SIGMA
7
,
033
(
2011
); e-print arXiv:1101.4469.
5.
R.
Askey
and
J.
Wilson
, “
Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
,”
Mem. Am. Math. Soc.
54
,
1
55
(
1985
).
6.
V. X.
Genest
,
H.
Miki
,
L.
Vinet
, and
A.
Zhedanov
, “
Spin lattices, state transfer and bivariate Krawtchouk polynomials
,”
Can. J. Phys.
93
,
979
984
(
2015
).
7.
M.
Christandl
,
N.
Datta
,
A.
Ekert
, and
A. J.
Landahl
, “
Perfect state transfer in quantum spin networks
,”
Phys. Rev. Lett.
92
,
187902
(
2004
); e-print arXiv:quant-ph/0309131.
8.
F.
Alberto Grunbaum
and
M.
Rahman
, “
A system of multivariable Krawtchouk polynomials and a probabilistic application
,”
SIGMA
7
,
119
136
(
2011
); e-print arXiv:1106.1835.
9.
S.
Odake
and
R.
Sasaki
, “
Solvable discrete quantum mechanics: q−orthogonal polynomials with |q| = 1 and quantum dilogarithm
,”
J. Math. Phys.
56
,
073502
(
2015
); e-print arXiv:1406.2768.
10.
I. G.
Macdonald
,
A New Class of Symmetric Functions
, Actes 20e Séminaire Lotharingen (
Publ. Inst. Rech. Math. Av.
,
Strasbourg, France
,
1988
), pp.
131
171
.
11.
I. G.
Macdonald
,
Affine Hecke Algebras and Orthogonal Polynomials
(
Cambridge University Press
,
2003
).
12.
T. H.
Koornwinder
, “
Askey-Wilson polynomials for root systems of type BC
,”
Contemp. Math.
138
,
189
204
(
1992
).
13.
J. V.
Stokman
, “
MacDonald-Koornwinder polynomials
,” e-print arXiv:1111.6112.
14.
S.
Sahi
, “
Nonsymmetric Koornwinder polynomials and duality
,”
Ann. Math.
150
,
267282
(
1999
); e-print arXiv:q-alg/9710032.
15.
S. N. M.
Ruijsenaars
, “
Complete integrability of relativistic Calogero-Moser systems and elliptic function identities
,”
Commun. Math. Phys.
175
,
75121
(
1995
).
16.
J.-F.
van Diejen
, “
Commuting difference operators with polynomial eigenfunctions
,”
Compos. Math.
95
,
183233
(
1995
); e-print arXiv:funct-an/9306002.
17.
J.-F.
van Diejen
, “
Difference Calogero-Moser systems and finite Toda chains
,”
J. Math. Phys.
36
,
1299
1323
(
1995
).
18.
M.
Feigin
and
A.
Silantyev
, “
Generalized MacDonald-Ruijsenaars systems
,”
Adv. Math.
250
,
144
192
(
2014
); e-print arXiv:1102.3903.
19.
J. F.
van Diejen
and
E.
Emsiz
, “
Integrable boundary interactions for Ruijsenaars’ difference Toda chain
,”
Commun. Math. Phys.
337
(
1
),
171
189
(
2015
).
20.
J. F.
van Diejen
and
E.
Emsiz
, “
Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit
,” e-print arXiv:1411.0463.
21.
J.-F.
van Diejen
, “
Diagonalization of an integrable discretization of the repulsive delta Bose gas on the circle
,”
Commun. Math. Phys.
267
,
451
476
(
2006
); e-print arXiv:math-ph/0604029.
22.
M.
Nazarov
and
E.
Sklyanin
, “
Lax operator for MacDonald symmetric functions
,”
Lett. Math. Phys.
105
,
901
916
(
2015
).
23.
M.
Nazarov
and
E.
Sklyanin
, “
Integrable hierarchy of the quantum Benjamin-Ono equation
,”
SIGMA
9
,
078
(
2013
); e-print arXiv:1309.6464.
24.
K.
Mimachi
and
Y.
Yamada
, “
Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials
,”
Commun. Math. Phys.
174
,
447
455
(
1995
).
25.
R.
Sakamoto
,
J.
Shiraishi
,
D.
Arnaudon
,
L.
Frappat
, and
E.
Ragoucy
, “
Correspondence between conformal field theory and Calogero-Sutherland model
,”
Nucl. Phys. B
704
,
490
509
(
2005
); e-print arXiv:hep-th/0407267.
26.
J. V.
Stokman
and
B. H. M.
Vlaar
, “
Koornwinder polynomials and the XXZ spin chain
,”
J. Approximation Theory
197
,
69
100
(
2015
).
27.
H. T.
Koelink
, “
Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications
,”
Acta Appl. Math.
44
,
295
352
(
1996
).
28.
A. S.
Zhedanov
, “
Hidden symmetry of Askey–Wilson polynomials
,”
Teoret. Mat. Fiz.
89
,
190204
(
1991
).
29.
D. B.
Fairlie
, “
Quantum deformations of SU(2)
,”
J. Phys. A: Math. Gen.
23
,
L183
(
1990
).
30.
M.
Havlícek
,
E.
Pelantová
, and
A.
Klimyk
, “
Nonstandard Uq(so(3)) and Uq(so(4)): Tensor products of representations, oscillator realizations and roots of unity
,”
Czech. J. Phys.
47
,
13
16
(
1996
).
31.

Note that the Askey-Wilson algebra first appeared in the context of Leonard pairs under the name “Leonard algebra.”See [Ref. 43, Definition 3.1].

32.
G.
Letzter
, “
Quantum zonal spherical functions and MacDonald polynomials
,”
Adv. Math.
189
,
88147
(
2004
); e-print arXiv:math/0210447.
33.
M.
Noumi
,
M. S.
Dijkhuizen
, and
T.
Sugitani
, “
Multivariable Askey-Wilson polynomials and quantum complex Grassmannians
,” in
Special functions, q−Series and Related Topics
, Fields Institute Communications (
American Mathematical Society
,
Providence, RI
,
1997
), Vol. 14, pp.
167
177
(Toronto, ON, 1995).
34.
I.
Cherednik
,
Double Affine Hecke Algebras
, London Mathematical Society Lecture Note Series (
Cambridge University Press
,
2005
), pp.
319
.
35.
I.
Cherednik
, “
Double affine Hecke algebras and MacDonald’s conjectures
,”
Ann. Math.
141
,
191216
(
1995
).
36.

Note that the two-variable case was constructed in Ref. 110.

37.
M. V.
Tratnik
, “
Multivariable Wilson polynomials
,”
J. Math. Phys.
30
,
2001
2011
(
1989
);
M. V.
Tratnik
Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials
,”
J. Math. Phys.
30
,
2740
2749
(
1989
);
M. V.
Tratnik
, “
Some multivariable orthogonal polynomials of the Askey tableau–continuous families
,”
J. Math. Phys.
32
,
2065
2073
(
1991
);
M. V.
Tratnik
, “
Some multivariable orthogonal polynomials of the Askey tableau-discrete families
,”
J. Math. Phys.
32
,
2337
2342
(
1991
).
38.
J.
Geronimo
and
P.
Iliev
, “
Bispectrality of multivariable Racah-Wilson polynomials
,”
Constr. Approx.
31
,
417
457
(
2010
); e-print arXiv:0705.1469.
39.
P.
Terwilliger
, “
Two relations that generalize the q−Serre relations and the Dolan-Grady relations
,” in
Proceedings of the Nagoya 1999 International Workshop on Physics and Combinatorics
, edited by
A. N.
Kirillov
,
A.
Tsuchiya
, and
H.
Umemura
(World Scientific Publishing,
1999
), pp.
377
398
; e-print arXiv:math.QA/0307016.
40.
P.
Baseilhac
, “
An integrable structure related with tridiagonal algebras
,”
Nucl. Phys. B
705
,
605
619
(
2005
); e-print arXiv:math-ph/0408025.
41.

For q≠1, see Ref. 110. For q = 1, two-variable Racah polynomials of Tratnik Ref. 37 arise in the representation theory of an extension of the Racah algebra QR (3), see Ref. 42.

42.
S.
Post
, “
Racah polynomials and recoupling schemes of su(1, 1)
,
SIGMA
11
,
057
(
2015
).
43.
P.
Terwilliger
, “
Leonard pairs and dual polynomial sequences
,”
1987
; avaiable at https://www.math.wisc.edu/∼terwilli/lphistory.html.
44.
P.
Terwilliger
, “
Leonard pairs and the q-Racah polynomials
,”
Linear Algebra Appl.
387
(
1
),
235
276
(
2004
).
45.
E. K.
Sklyanin
, “
Boundary conditions for integrable quantum systems
,”
J. Phys. A
21
,
2375
2389
(
1988
).
46.
L.
Onsager
, “
Crystal statistics. I. A two-dimensional model with an order-disorder transition
,”
Phys. Rev.
65
,
117
149
(
1944
).
47.
B.
Davies
, “
Onsager’s algebra and superintegrability
,”
J. Phys. A
23
,
2245
2261
(
1990
);
B.
Davies
, “
Onsager’s algebra and the Dolan-Grady condition in the non-self-dual case
,”
J. Math. Phys.
32
,
2945
2950
(
1991
).
48.
G.
von Gehlen
and
V.
Rittenberg
, “
Zn-symmetric quantum chains with infinite set of conserved charges and Zn zero modes
,”
Nucl. Phys. B
257
,
351
(
1985
).
49.
H.
Araki
, “
Master symmetries of the XY model
,”
Commun. Math. Phys.
132
,
155
176
(
1990
).
50.
C.
Ahn
and
K.
Shigemoto
, “
Onsager algebra and integrable lattice models
,”
Mod. Phys. Lett. A
6
,
3509
(
1991
).
51.

For triangular, diagonal, or special Uq(sl2) invariant boundary conditions, quotients of the q −Onsager algebra have to be considered.

52.
P.
Baseilhac
and
S.
Belliard
, “
Generalized q-Onsager algebras and boundary affine Toda field theories
,”
Lett. Math. Phys.
93
,
213
228
(
2010
); e-print arXiv:0906.1215.
53.
P.
Baseilhac
and
K.
Koizumi
, “
A new (in)finite dimensional algebra for quantum integrable models
,”
Nucl. Phys. B
720
,
325
347
(
2005
); e-print arXiv:math-ph/0503036.
54.

For the Ising or superintegrable Potts model (q = 1), hk = δk0, h0=0 with Kronecker symbol δkl. For the open XXZ chain, the explicit expressions for hk and h0 are given in Ref. 59.

55.

The parameters ω0, ω1, g+, g− are model-dependent. For the Ising and superintegrable chiral Potts model (q = 1), ω0 = 1, ω1 is the free magnetic field coupled with the system. For the open XXZ chain, ω0, ω1, g+, g− are associated with the boundary parameters.59 

56.
P.
Baseilhac
and
K.
Shigechi
, “
A new current algebra and the reflection equation
,”
Lett. Math. Phys.
92
,
47
65
(
2010
); e-print arXiv:0906.1482.
57.
P.
Baseilhac
and
S.
Belliard
, “
A note on the O q ( s l 2 ^ ) algebra
,” e-print arXiv:1012.5261.
58.
P.
Baseilhac
and
S.
Belliard
, “
The half-infinite XXZ chain in Onsager’s approach
,”
Nucl. Phys. B
873
,
550
583
(
2013
); e-print arXiv:1211.6304.
59.
P.
Baseilhac
and
K.
Koizumi
, “
A deformed analogue of Onsager’s symmetry in the XXZ open spin chain
,”
J. Stat. Mech.
,
P10005
(
2005
); e-print arXiv:hep-th/0507053;
P.
Baseilhac
and
K.
Koizumi
, “
Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
,”
J. Stat. Mech.
,
P09006
(
2007
); e-print arXiv:hep-th/0703106.
60.
G.
Gasper
and
M.
Rahman
,
Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications
, 2nd ed. (
Cambridge Universiy Press
,
2004
), Vol. 96.
61.

Compared to the notation Eq,zj in Ref. 2, one has E¯zjEq2,zj.

62.
P.
Terwilliger
, “
The subconstituent algebra of an association scheme. III
,”
J. Algebraic Combin.
2
,
177
210
(
1993
).
63.
T.
Ito
,
K.
Tanabe
, and
P.
Terwilliger
, “
Some algebra related to P- and Q-polynomial association schemes, codes and association schemes
,” in
DIMACS Series in Discrete Mathematics and Theoretical Computer Science
(
American Mathematical Society
,
Providence, RI
,
2001
), Vol. 56, pp.
167
192
(Piscataway, NJ, 1999); e-print arXiv:math/0406556v1.
64.
L.
Dolan
and
M.
Grady
, “
Conserved charges from self-duality
,”
Phys. Rev. D
25
,
1587
1604
(
1982
).
65.

The q−commutator X,Yq=qXYq1YX, where q is the deformation parameter, is introduced.

66.
Ya. I.
Granovskii
,
I. M.
Lutzenko
, and
A. S.
Zhedanov
, “
Mutual integrability, quadratic alge- bras, and dynamical symmetry
,”
Ann. Phys.
217
,
120
(
1992
).
67.
F. A.
Grunbaum
and
L.
Haine
, “
The q−version of a theorem of Bochner
,”
J. Comput. Appl. Math.
68
,
103
114
(
1996
).
68.
M.
Noumi
and
J.
Stokman
, “
Askey-Wilson polynomials: An affine Hecke algebraic approach
,” Advances in the theory of special functions and orthogonal polynomials, Proceedings title: Laredo Lectures on Orthogonal Polynomials and Special Functions, p.
111
144
.
69.

If a > 1, b, c, d are real or one is real and the other two are complex conjugates, max(|b|, |c|, |d|) < 1 and the pairwise products of ab, ac, ad, bc, bd have modulus less than one, then the polynomials satisfy a different orthogonality relation [Ref. 70, Eq. (3.1.3)].

70.
R.
Koekoek
and
R.
Swarttouw
, “
The Askey-scheme of hypergeometric orthogonal polynomials and its q−analogue
,” e-print arXiv:math.CA/9602214v1.
71.

Then, qnj determines uniquely nj.

72.
A.
Kirillov
and
M.
Noumi
, “
Affine Hecke algebras and raising operators for Macdonald polynomials
,”
Duke Math. J.
93
,
1
39
(
1998
); e-print arXiv:q-alg/9605004.
73.
E.
Bannai
and
T.
Ito
,
Algebraic Combinatorics I: Association Schemes
(
Benjamin/Cummings Publishing Company
,
1984
), p.
260
.
74.
P.
Terwilliger
, “
Two linear transformations each tridiagonal with respect to an eigenbasis of the other
,”
Linear Algebra Appl.
330
,
149
203
(
2001
); e-print arXiv:math.RA/0406555.
75.
T.
Ito
and
P.
Terwilliger
, “
The augmented tridiagonal algebra
,”
Kyushu J. Math.
64
,
81
144
(
2010
); e-print arXiv:0904.2889.
76.
T.
Ito
,
K.
Nomura
, and
P.
Terwilliger
, “
A classification of sharp tridiagonal pairs
,”
Linear Algebra Appl.
435
,
1857
1884
(
2011
); e-print arXiv:1001.1812.
77.
T.
Ito
and
P.
Terwilliger
, “
Tridiagonal pairs of q−Racah type
,”
J. Algebra
322
,
6893
(
2009
); e-print arXiv:0807.0271.
78.

For applications to quantum integrable systems generated by the q−Onsager algebra, scalar products of the Hamiltonian’s eigenstates will be expressed in terms of the overlap coefficients.

79.
P.
Baseilhac
, “
A family of tridiagonal pairs and related symmetric functions
,”
J. Phys. A
39
,
11773
(
2006
); e-print arXiv:math-ph/0604035.
80.
K.
Nomura
and
P.
Terwilliger
, “
Tridiagonal pairs of q-Racah type and the μ-conjecture
,”
Linear Algebra Appl.
432
,
615
636
(
2010
); e-print arXiv:0908.3151.
81.
P.
Terwilliger
, “
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; Comments on the parameter array
,”
Designs, Codes Cryptography
34
,
307
332
(
2005
); e-print arXiv:math/0306291.
82.

By construction, note that the coefficient c0 = 0.

83.

Note that a different embedding can be considered; see Ref. 79, Proposition 2.2.

84.

We especially thank Paul Terwilliger for pointing out Proposition 1.24 and other results of Ref. 75 to us.

85.
P.
Baseilhac
and
T.
Kojima
, “
Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
,”
Nucl. Phys. B
880
,
378
413
(
2014
); e-print arXiv:1309.7785;
P.
Baseilhac
and
T.
Kojima
, “
Form factors of the half-infinite XXZ spin chain with a triangular boundary
,”
J. Stat. Mech.
,
P09004
(
2014
); e-print arXiv:1404.0491.
86.

It is the case for the open XXZ spin chain; see Ref. 59.

87.
S.
Faldella
,
N.
Kitanine
, and
G.
Niccoli
, “
Complete spectrum and scalar products for the open spin-1/2 XXZ quantum chains with non-diagonal boundary terms
,”
J. Stat. Mech.
P01011
(
2014
).
88.

In the literature, the notation rn(λ(x); a, b, c, d) with λ(x) = x(x + c+ d + 1) is sometimes used Ref. 70.

89.
J.
Cao
,
H.-Q.
Lin
,
K.-J.
Shi
, and
Y.
Wang
, “
Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields
,”
Nucl. Phys. B
663
,
487
(
2003
); e-print arXiv:cond-mat/0212163.
90.
W.-L.
Yang
and
Y.-Z.
Zhang
, “
On the second reference state and complete eigenstates of the open XXZ chain
,”
JHEP
04
,
044
(
2007
); e-print arXiv:hep-th/0703222.
91.
G.
Filali
and
N.
Kitanine
, “
Spin chains with non-diagonal boundaries and trigonometric SOS model with reflecting end
,”
SIGMA
7
,
1
22
(
2011
).
92.
S.
Belliard
, “
Modified algebraic Bethe ansatz for XXZ chain on the segment. I. Triangular cases
,”
Nucl. Phys. B
892
,
1
20
(
2015
); e-print arXiv:1408.4840.
93.
S.
Belliard
and
R. A.
Pimenta
, “
Modified algebraic Bethe ansatz for XXZ chain on the segment. II. General cases
,”
Nucl. Phys. B
894
(
5
),
527
552
(
2015
).
94.
N.
Crampé
, “
Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
,”
J. Phys. A: Math. Theor.
48
,
08FT01
(
2015
); e-print arXiv:1411.7954.
95.
X.
Zhang
,
Y.-Y.
Li
,
J.
Cao
,
W.-L.
Yang
,
K.
Shi
, and
Y.
Wang
, “
Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
,”
Nucl. Phys. B
893
,
70
88
(
2015
); e-print arXiv:1412.6905v2.
96.
W.
Galleas
, “
Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions
,” e-print arXiv:0708.0009.
97.
W.
Galleas
, “
Off-shell scalar products for the XXZ spin chain with open boundaries
,”
Nucl. Phys. B
893
,
346
375
(
2015
); e-print arXiv:1412.5389.
98.
R. I.
Nepomechie
, “
Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms
,”
J. Phys. A
37
,
433
440
(
2004
); e-print arXiv:hep-th/0304092.
99.
J.
Cao
,
W.-L.
Yang
,
K.
Shi
, and
Y.
Wang
, “
Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields
,”
Nucl. Phys. B
877
,
152
175
(
2013
);
J.
Cao
,
W.-L.
Yang
,
K.
Shi
, and
Y.
Wang
,
J. Phys. A: Math. Theor.
48
(
44
) (
2015
);
J.
Cao
,
W.-L.
Yang
,
K.
Shi
, and
Y.
Wang
, “
On the complete-spectrum characterization of quantum integrable spin chains via the inhomogeneous T-Q relation
,” e-print arXiv:1409.5303.
100.
N.
Kitanine
,
J.-M.
Maillet
, and
G.
Niccoli
, “
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
,”
J. Stat. Mech.
P05015
(
2014
).
101.

Recall that zeroes of Askey-Wilson polynomials satisfy Bethe equations associated with a “1”-site open XXZ chain, as pointed out in Ref. 102.

102.
P. B.
Wiegmann
and
A. V.
Zabrodin
, “
Algebraization of difference eigenvalue equations related to Uq(sl 2)
,”
Nucl. Phys. B
451
,
699
(
1995
); e-print arXiv:cond-mat/9501129.
103.
N.
Kitanine
,
K. K.
Kozlowski
,
J. M.
Maillet
,
G.
Niccoli
,
N. A.
Slavnov
, and
V.
Terras
, “
Correlation functions of the open XXZ chain I
,”
JSTAT
,
P10009
(
2007
); e-print arXiv:0707.1995;
N.
Kitanine
,
K. K.
Kozlowski
,
J. M.
Maillet
,
G.
Niccoli
,
N. A.
Slavnov
, and
V.
Terras
Correlation functions of the open XXZ chain II
,”
JSTAT
P07010
(
2008
); e-print arXiv:0803.3305.
104.
M.
Jimbo
,
R.
Kedem
,
T.
Kojima
,
H.
Konno
, and
T.
Miwa
, “
XXZ chain with a boundary
,”
Nucl. Phys. B
441
,
437
470
(
1995
); e-print arXiv:hep-th/9411112;
M.
Jimbo
,
R.
Kedem
,
H.
Konno
,
T.
Miwa
and
R.
Weston
, “
Difference equations in spin chains with a boundary
,”
Nucl. Phys. B
448
,
429
456
(
1995
); e-print arXiv:hep-th/9502060.
105.
S.
Kolb
, “
Quantum symmetric Kac-Moody pairs
,”
Adv. Math.
267
,
395
469
(
2014
); e-print arXiv:1207.6036v1.
106.
D.
Uglov
and
L.
Ivanov
, “
(N) Onsager’ s algebra and integrability
,”
J. Stat. Phys.
82
,
87
(
1996
); e-print arXiv:hep-th/9502068v1.
107.
F. A.
Grunbaum
and
L.
Haine
, “
A theorem of Bochner, revisited
,” in
Algebraic Aspects of Integrable Systems
, Progress in Nonlinear Differential Equations, edited by
A. S.
Fokas
and
I. M.
Gel’fand
(
Birkhauser
,
Boston
,
1996
), Vol. 26, p.
143172
, in memory of I. Dorfman.
108.
F. A.
Grunbaum
and
L.
Haine
, “
Some functions that generalize the Askey-Wilson polynomials
,”
Commun. Math. Phys.
184
,
173202
(
1997
).
109.
J.
Geronimo
and
P.
Iliev
, “
Multivariable Askey-Wilson function and bispectrality
,”
Ramanujan J.
24
,
273
287
(
2011
).
110.
H. T.
Koelink
and
J.
Van Der Jeugt
, “
Convolutions for orthogonal polynomials from Lie and quantum algebra representations
,”
SIAM J. Math. Anal.
29
,
794
822
(
1998
); e-print arXiv:q-alg/9607010.
You do not currently have access to this content.