The Heun–Racah and Heun–Bannai–Ito algebras are introduced. Specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic Heun construct to the bispectral operators of the Racah and Bannai–Ito polynomials. The study supplements the results on the Heun–Askey–Wilson algebra and completes the description of the Heun algebras associated with the polynomial families at the top of the Askey scheme, its q-analog, and the Bannai–Ito one.
REFERENCES
1.
F. A.
Grünbaum
, L.
Vinet
, and A.
Zhedanov
, “Algebraic Heun operator and band-time limiting
,” Commun. Math. Phys.
364
(3
), 1041
–1068
(2018
).2.
F. A.
Grünbaum
. The Bispectral Problem: An Overview
(Springer Netherlands
, Dordrecht
, 2001
), pp. 129
–140
.3.
N.
Crampé
, L.
Vinet
, and A.
Zhedanov
, “Heun algebras of Lie type
,” Proc. Am. Math. Soc.
148
, 1079
–1094
(2020
).4.
D.
Slepian
, “Some comments on Fourier analysis, uncertainty and modeling
,” SIAM Rev.
25
(3
), 379
–393
(1983
).5.
H. J.
Landau
, “An overview of time and frequency limiting
,” in Fourier Techniques and Applications
, edited by J. F.
Price
(Plenum
, New York
, 1985
), pp. 201
–220
.6.
N.
Crampé
, R. I.
Nepomechie
, and L.
Vinet
, “Free-fermion entanglement and orthogonal polynomials
,” J. Stat. Mech.: Theory Exp.
2019
(9
), 093101
.7.
N.
Crampé
, R. I.
Nepomechie
, and L.
Vinet
, “Entanglement in fermionic chains and bispectrality
,” in (World Scientific
, 2020
), Chap. 13, pp. 77
–96
.8.
R.
Koekoek
, P. A.
Lesky
, and R. F.
Swarttouw
, Hypergeometric Orthogonal Polynomials and Their q-Analogues
, Springer Monographs in Mathematics (Springer
, 2010
).9.
F. A.
Grünbaum
, L.
Vinet
, and A.
Zhedanov
, “Tridiagonalization and the Heun equation
,” J. Math. Phys.
58
(3
), 1
–17
(2017
).10.
L.
Vinet
and A.
Zhedanov
, “The Heun operator of Hahn-type
,” Proc. Am. Math. Soc.
147
, 2987
(2019
).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
, 367
–380
(2020
).12.
K.
Takemura
, “On q-deformations of the Heun equation
,” Symmetry, Integrability Geom.: Methods Appl.
14
, 061
(2018
).13.
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
, 3091
–3112
(2019
).14.
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
).15.
S.
Belliard
and N.
Crampé
, “Heisenberg XXX model with general boundaries: Eigenvectors from algebraic Bethe ansatz
,” Symmetry, Integrability Geom.: Methods Appl.
9
, 072
(2013
).16.
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
).17.
K.
Nomura
and P.
Terwilliger
, “Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair
,” Linear Algebra Appl.
420
(1
), 198
–207
(2007
).18.
M. E. H.
Ismail
and E.
Koelink
, “Spectral properties of operators using tridiagonalization
,” Analysis Appl.
10
(03
), 327
–343
(2012
).19.
V. X.
Genest
, M. E. H.
Ismail
, L.
Vinet
, and A.
Zhedanov
, “Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra
,” Proc. Am. Math. Soc.
144
(10
), 4441
–4454
(2016
).20.
V. X.
Genest
, L.
Vinet
, and A.
Zhedanov
, “The Racah algebra and superintegrable models
,” J. Phys.: Conf. Ser.
512
(1
), 012011
(2014
).21.
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
).22.
V. X.
Genest
, L.
Vinet
, and A.
Zhedanov
, “Embeddings of the Racah algebra into the Bannai-Ito algebra
,” Symmetry, Integrability Geom.: Methods Appl.
11
, 050
(2015
).23.
H.
De Bie
, V. X.
Genest
, S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “The Bannai-Ito algebra and some applications
,” J. Phys.: Conf. Ser.
597
(1
), 012001
(2015
).24.
S.
Tsujimoto
, L.
Vinet
, and A.
Zhedanov
, “Dunkl shift operators and Bannai–Ito polynomials
,” Adv. Math.
229
(4
), 2123
–2158
(2012
).25.
26.
L.
Vinet
and A.
Zhedanov
, “A “missing” family of classical orthogonal polynomials
,” J. Phys. A: Math. Theor.
44
(8
), 085201
(2010
).27.
V. X.
Genest
, L.
Vinet
, and A.
Zhedanov
, “Bispectrality of the complementary Bannai-Ito polynomials
,” Symmetry, Integrability Geom.: Methods Appl.
9
, 018
(2013
).© 2020 Author(s).
2020
Author(s)
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