The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second-order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.
REFERENCES
1.
E. K.
Sklyanin
, “Some algebraic structures connected with the Yang–Baxter equation. Representations of quantum algebras
,” Funct. Anal. Appl.
17
(4
), 273
–284
(1984
).2.
S. P.
Smith
, “The four-dimensional Sklyanin algebras
,” K-Theory
8
(1
), 65
–80
(1994
).3.
C.
Walton
, “Representation theory of three-dimensional Sklyanin algebras
,” Nucl. Phys. B
860
(1
), 167
–185
(2012
).4.
N.
Iyudu
and S.
Shkarin
, “Three dimensional Sklyanin algebras and Gröbner bases
,” J. Algebra
470
, 379
–419
(2017
).5.
F. A.
Grünbaum
, L.
Vinet
, and A.
Zhedanov
, “Algebraic Heun operator and band-time limiting
,” Commun. Math. Phys.
364
(3
), 1041
–1068
(2018
).6.
K.
Takemura
, “On q-deformations of the Heun equation
,” SIGMA
14
(061
), 16
(2018
).7.
P.
Baseilhac
, S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “The Heun–Askey–Wilson algebra and the Heun operator of Askey–Wilson type
,” Ann. Henri Poincaré
20
(9
), 3091
–3112
(2019
).8.
L.
Vinet
and A.
Zhedanov
, “The Heun operator of Hahn-type
,” Proc. Am. Math. Soc.
147
(7
), 2987
–2998
(2019
).9.
N.
Crampé
, L.
Vinet
, and A.
Zhedanov
, “Heun algebras of Lie type
,” Proc. Am. Math. Soc.
148
(3
), 1079
–1094
(2019
).10.
S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “The rational Heun operator and Wilson biorthogonal functions
,” Ramanujan J.
(published online 2021
).11.
P.
Baseilhac
, L.
Vinet
, and A.
Zhedanov
, “The q-Heun operator of big q-Jacobi type and the q-Heun algebra
,” Ramanujan J.
52
(2
), 367
–380
(2020
).12.
G.
Bergeron
, N.
Crampé
, S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “The Heun–Racah and Heun–Bannai–Ito algebras
,” J. Math. Phys.
61
(8
), 081701
(2020
).13.
D.
Slepian
, “Some comments on Fourier analysis, uncertainty and modeling
,” SIAM Rev.
25
(3
), 379
–393
(1983
).14.
H. J.
Landau
, “An overview of time and frequency limiting
,” in Fourier Techniques and Applications
, edited by J. F.
Price
(Springer
, Boston, MA
, 1985
), pp. 201
–220
.15.
G.
Bergeron
, L.
Vinet
, and A.
Zhedanov
, “Signal processing, orthogonal polynomials, and Heun equations
,” in Orthogonal Polynomials
, Tutorials, Schools, and Workshops in the Mathematical Sciences, edited by M.
Foupouagnigni
and W.
Koepf
(Springer International Publishing
, Cham
, 2020
), pp. 195
–214
.16.
N.
Crampé
, R. I.
Nepomechie
, and L.
Vinet
, “Entanglement in fermionic chains and bispectrality
,” Rev. Math. Phys.
33
(07
), 2140001
(2021
).17.
N.
Crampé
, R. I.
Nepomechie
, and L.
Vinet
, “Free-Fermion entanglement and orthogonal polynomials
,” J. Stat. Mech.: Theory Exp.
2019
(9
), 093101
.18.
S.
Belliard
and R. A.
Pimenta
, “Modified algebraic Bethe ansatz for XXZ chain on the segment—II—General cases
,” Nucl. Phys. B
894
, 527
–552
(2015
).19.
P.
Baseilhac
and R. A.
Pimenta
, “Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
,” Nucl. Phys. B
949
, 114824
(2019
).20.
J.
Gaboriaud
, S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators
,” J. Phys. A: Math. Theor.
53
(44
), 445204
(2020
).21.
A. S.
Gorsky
and A. V.
Zabrodin
, “Degenerations of Sklyanin algebra and Askey-Wilson polynomials
,” J. Phys. A: Math. Gen.
26
(15
), L635
–L639
(1993
).22.
E. G.
Kalnins
and W.
Miller
, “Symmetry techniques for q-series: Askey-Wilson polynomials
,” Rocky Mt. J. Math.
19
(1
), 223
–230
(1989
).23.
A. S.
Zhedanov
, “‘Hidden symmetry’ of Askey-Wilson polynomials
,” Theor. Math. Phys.
89
(2
), 1146
–1157
(1991
).24.
G.
Bergeron
, J.
Gaboriaud
, L.
Vinet
, and A.
Zhedanov
, “Sklyanin-like algebras for (q-)linear grids and (q-)para-Krawtchouk polynomials
,” J. Math. Phys.
62
(1
), 013505
(2021
).25.
P. B.
Wiegmann
and A. X.
Zabrodin
, “Algebraization of difference eigenvalue equations related to Uq(sl2)
,” Nucl. Phys. B
451
(3
), 699
–724
(1995
).26.
A.
Smirnov
, “Degenerate Sklyanin algebras
,” Cent. Eur. J. Phys.
8
(4
), 542
–554
(2010
).27.
W.
Miller
, Jr., “A note on Wilson polynomials
,” SIAM J. Math. Anal.
18
(5
), 1221
–1226
(1987
).28.
J.-M.
Lemay
, L.
Vinet
, and A.
Zhedanov
, “The para-Racah polynomials
,” J. Math. Anal. Appl.
438
(2
), 565
–577
(2016
).29.
R.
Koekoek
, P. A.
Lesky
, and R. F.
Swarttouw
, Hypergeometric Orthogonal Polynomials and Their Q-Analogues
, Springer Monographs in Mathematics (Springer
, Berlin, Heidelberg
, 2010
).30.
L.
Vinet
and A.
Zhedanov
, “Generalized Bochner theorem: Characterization of the Askey–Wilson polynomials
,” J. Comput. Appl. Math.
211
(1
), 45
–56
(2008
).31.
Y. A.
Granovskii
and A. S.
Zhedanov
, “Nature of the symmetry group of the 6j-symbol
,” J. Exp. Theor. Phys.
67
, 1982
–1985
(1988
).32.
J. S.
Geronimo
and P.
Iliev
, “Bispectrality of multivariable Racah–Wilson polynomials
,” Constr. Approximation
31
(3
), 417
–457
(2010
).33.
V. X.
Genest
, L.
Vinet
, and A.
Zhedanov
, “Superintegrability in two dimensions and the Racah–Wilson algebra
,” Lett. Math. Phys.
104
(8
), 931
–952
(2014
).34.
F. A.
Grünbaum
, L.
Vinet
, and A.
Zhedanov
, “Tridiagonalization and the Heun equation
,” J. Math. Phys.
58
(3
), 031703
(2017
).35.
E. M.
Rains
, “BCn-symmetric abelian functions
,” Duke Math. J.
135
(1
), 99
–180
(2006
).36.
L.
Vinet
and A.
Zhedanov
, “Para-Krawtchouk polynomials on a bi-lattice and a quantum spin chain with perfect state transfer
,” J. Phys. A: Math. Theor.
45
, 265304
(2012
); arXiv:1110.6475.© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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