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The quantum loop algebra of $sl_2$ and $q$-Racah type bivariate functions

T0 review · 5 major / 8 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper shows that two six-parameter families of bivariate q-Racah functions are overlap coefficients between six eigenbases built from the quantum loop algebra, unifying them under one algebraic framework.

desk verdict A solid, explicit construction of bivariate q-Racah overlaps from LU_q(sl2) representations, with a TD-pair characterization that needs one displayed parameter-transfer check. read the letter →

arxiv 2607.20043 v1 pith:IYTPBUZN submitted 2026-07-22 math.QA math-phmath.MPmath.RT

Pascal Baseilhac , Nicolas Crampe This is my paper · ORCID
classification math.QAmath-phmath.MPmath.RT MSC 16T1016G6033C80
keywords quantumloopalgebraq-RacahpolynomialsbivariateorthogonaltridiagonalpairsfactorizedLeonardAskey–Wilsonq-Onsagercoidealsubalgebras
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish a single algebraic origin for two different six-parameter families of bivariate q-Racah type functions, which are multivariate analogues of the classical q-Racah orthogonal polynomials. The authors construct six 'distinguished' eigenbases of tensor-product evaluation representations of the quantum loop algebra of sl_2, labeled by two evaluation parameters, and show that the overlap coefficients between these bases are exactly the two families, denoted S and T. If correct, this gives a uniform representation-theoretic explanation for the bivariate q-Racah functions, links them to tridiagonal pairs and factorized Leonard pairs, and opens a path toward multivariate q-Racah functions. The central mechanism is a set of eight elements in the tensor square of the quantum loop algebra, built from a coideal subalgebra embedding of the q-Onsager algebra, which yield six commuting pairs of operators whose joint eigenbases produce the desired functions.

What carries the argument

The central object is the set of eight elements in L U_q sl_2 ⊗ L U_q sl_2, built from two left/right coideal subalgebras (images of the q-Onsager algebra under a parameter-dependent embedding) and the coproduct. From these eight elements, six commutative pairs are formed, and their simultaneous eigenbases on tensor-product evaluation representations yield the bivariate functions. The key identity is the explicit overlap formula expressing products of q-Racah polynomials as coefficients relating these eigenbases. The TD-pair characterization relies on a parameter correspondence with the Itô–Terwilliger irreducibility conditions, and the bispectrality at u1 = u2 is proved via a homomorphism f

What would settle it

Compute the overlap coefficients S and T for the smallest nontrivial cases (e.g., j1 = 1/2, j2 = 1) and check that they satisfy the claimed TD-pair block-tridiagonal actions with the stated diameter and character; also verify the parameter correspondence (4.4) by explicitly checking irreducibility of the q-Onsager module for a few parameter values.

Watch

Extended reading notes

Core claim

The central claim is that the bivariate q-Racah type functions S and T are overlap coefficients relating six distinguished eigenbases of the tensor product of two evaluation representations of L U_q sl_2, parametrized by scalars a, b and evaluation parameters u1, u2. Concretely, Theorem 3.7 gives explicit formulas for these overlaps as products of two univariate q-Racah polynomials. Under generic conditions, the pairs (A12, B21) and (A21, B12) act as tridiagonal pairs of q-Racah type (Propositions 4.3 and 4.5), and the S-overlaps are the corresponding overlap coefficients. When u1 = u2, the T-overlaps become bispectral bivariate polynomials, related to the rank 2 Askey–Wilson algebra via a h

Load-bearing premise

The proof that the pairs (A12, B21) and (A21, B12) are tridiagonal pairs relies on an asserted parameter correspondence with the Itô–Terwilliger irreducibility theorem, but the full equivalence is not displayed; if that correspondence misses a normalization or convention, the 'if and only if' TD-pair claim is unsupported.

Editorial extensions

If this is right

  • The two six-parameter families of bivariate q-Racah functions are shown to arise from a single representation-theoretic framework, rather than as separate ad hoc constructions.
  • Under the stated conditions, the S-functions are overlap coefficients for tridiagonal pairs of q-Racah type, with explicit diameter, shape, and character formulas.
  • At equal evaluation parameters, the T-functions are bispectral bivariate polynomials and the algebraic structure is governed by the rank 2 Askey–Wilson algebra.
  • If Conjecture 4.20 holds, the T-functions give concrete overlap coefficients for factorized Leonard pairs of type B'_2, extending the known A_2 case.
  • The construction suggests a route toward multivariate (N > 2) q-Racah type functions with 2N+2 parameters, as the authors conjecture in the conclusion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct consequence of the overlap interpretation is that the bispectrality and orthogonality of these bivariate functions can be derived from the commutation relations of the eight elements, bypassing case-by-case hypergeometric proofs; this is a testable extension that the paper only partially pursues.
  • The failure of the block recurrence and difference equations to uniquely determine S suggests that a second commuting pair (e.g., from alternating generators) is needed; such a pair, if found, would provide a complete characterization and could also resolve the q → 1 discrepancy with the Racah-type overlaps.
  • The B'_2 factorized Leonard pair conjecture implies a root-system interpretation of the u1 = u2 specialization, which may connect to higher-rank Askey–Wilson algebras and to multivariate Griffiths polynomials; the authors mention Griffiths polynomials in the concluding remarks as a natural further domain.
  • The parameter correspondence (4.4) used to import irreducibility conditions is the most delicate step; checking it against explicit small-dimensional representations (j1, j2 small) would either validate or refute the claimed 'if and only if' TD-pair result.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 8 minor

Summary. The paper constructs six distinguished eigenbases of tensor-product evaluation representations of the quantum loop algebra L U_q sl_2, labelled by two evaluation parameters u_1,u_2 and two scalars a,b. The overlap coefficients between these bases are computed in Theorem 3.7 as products of two univariate q-Racah polynomials (the S and T families). For the S-overlaps, the pairs (A_12,B_21) and (A_21,B_12) are asserted to be TD pairs of q-Racah type under explicit conditions (Prop. 4.3), with diameter, shape, character, and overlap identification (Props. 4.5, 4.8). For u_1=u_2, the T-overlaps are shown to enjoy bispectrality via four relations (Thm. 4.12), the bispectral algebra is identified with rank-two Askey-Wilson algebra AW(4) (Prop. 4.14), and a new notion of factorized B'_2-Leonard pair is introduced with a conjecture that (A_2,A_12;B_12,B_1) forms one (Conj. 4.20).

Significance. If the structural claims hold, this gives a genuinely higher-rank algebraic interpretation of two six-parameter families of bivariate q-Racah-type functions: the S-family is connected to tridiagonal pairs of q-Racah type, answering an explicit question of Terwilliger, and the T-family at u_1=u_2 is connected to AW(4) and conjecturally to a new factorized-Leonard-pair structure. The core overlap computation (Theorem 3.7) is explicit, self-contained, and reduces to the univariate Rosengren formula, so the pairwise-eigenbasis part is solid and falsifiable. The paper is also commendably candid: it flags its conjectural parts (Conj. 4.20), its unshown computer check (Prop. 4.11), and the mismatch with [CGT25] in the q→1 limit. The main risk is not circularity (no fitted constants; univariate q-Racah formulas from [R03] are independent support) but the correctness of the parameter-transfer in the TD-pair characterization and the completeness of the displayed proof.

major comments (5)
  1. [§4.1, Prop. 4.3] The 'if and only if' TD-pair characterization is load-bearing for the advertised structural interpretation of the S-overlaps, but its proof rests entirely on an unshown substitution of (4.4) into [IT09, Thm. 1.17(i)]. The text states the comparison of split-basis actions and the normalization coefficients (4.5)-(4.6), but it does not display the actual substitution or the verification that the resulting conditions are identical to (i)-(iii). Because the q-Onsager embedding here differs from the one in [IT09] (Remarks 2.5 and 3.1), a sign, q-power, or tensor-factor-ordering error could make conditions (i)-(iii) not exactly the irreducibility conditions for (A_12,B_21). Please either reproduce the full substitution with each of (i1)-(i3) verified, or state Prop. 4.3 as a conditional statement with the transfer as an explicit lemma. A direct irreducibility test on small j_1,j_2 would also s
  2. [§4.1, eqs. (4.5)-(4.6)] The normalization coefficients c_{n1,n2} relating the split basis to [I13]'s basis are asserted with only two recurrence constraints displayed. The closed form (4.6) is claimed as a solution, but no consistency check (e.g., path independence around the rectangle) is shown, and the exact relation to [I13]'s normalization is not derived. Since the TD-pair proof relies on exact identification with [IT09]'s irreducibility theorem, a small normalization mismatch would break the 'iff'. Either derive (4.6) fully or make explicit that (4.3) is an ansatz needing independent verification.
  3. [§4.2.1, Prop. 4.11] Proposition 4.11 states that for u_1=u_2 the actions (4.17)-(4.18) have support on B'_2, but the proof says only that a computer-algebra check was performed and the explicit coefficients are omitted. This is load-bearing for Theorem 4.12's four-term bispectrality relations and for the B'_2-Leonard structure. Please include the actual coefficient expressions, or at minimum provide a reproducible symbolic script with the q-Pochhammer identities used, so the check can be verified by a referee.
  4. [§4.2.4, Defs. 4.17-4.18 and Conj. 4.20] The FL-pair connection is explicitly conjectural, which is acceptable. However, the new definitions of B'_2-Leonard pair and factorized B'_2-Leonard pair introduce axioms (i)-(xi) whose consistency and non-vacuity are not demonstrated. Since Conjecture 4.20 asserts that (A_2,A_12;B_12,B_1) satisfies these axioms, the paper would be strengthened by verifying the dimension-one, irreducibility, and orthogonality conditions for a small example (e.g., j_1=j_2=1/2) under the stated generic conditions.
  5. [§4.2.2, Prop. 4.14] The homomorphism ψ: AW(4) → U_q sl_2 ⊗ U_q sl_2 is asserted with the proof summarized as 'verification of all the defining relations'. This is a key structural result connecting the T-overlaps to AW(4). Please provide a representative derivation of one nontrivial defining relation (e.g., (4.27) for (I,J,K)=(1,2,3)) or give a systematic computer-check protocol, so the reader can assess the conventions, especially the images of G_3 and G_4, which appear to be scalars.
minor comments (8)
  1. [§3.3] The genericity conditions for V_{2j1}(u1)⊗V_{2j2}(u2) to be irreducible are not stated precisely; the text says 'Unless ratios... are certain powers of q' with reference to [CP91]. State the exact condition on u1/u2.
  2. [§2.3, Eq. (2.28)] The q-Racah formula is quoted as a straightforward application of [R03, Eq. (2.16)]. A brief derivation or appendix checking the pre-factor would make the paper more self-contained.
  3. [§2.3, Eqs. (2.31)-(2.35)] The substitution notation 'a↔b' and 'b→bq^{2n1-2j1}, u→u2' is terse. State the full functional dependence of the coefficients A_{i,j}, B_{i,j} explicitly.
  4. [Figure 1] The caption does not explain the dashed/doubled/simple arcs. Add a caption describing which arcs correspond to which S/T overlap coefficients.
  5. [§4.1, Remark 4.4] The list of nine terms is ambiguous; specify all sign combinations in the index set (e.g., u_{n1±1,n2∓1} means four combinations).
  6. [§4.2.1, Eqs. (4.12)-(4.13)] The '∼' notation for 'up to functions f(k_1,k_2) and g(ℓ_1,ℓ_2)' should define these functions or state that they are not needed explicitly.
  7. [§5] The discussion of Griffiths polynomials is speculative; either shorten it or state an explicit limiting connection.
  8. [Notation] The q-commutator convention [X,Y]_q = qXY - q^{-1}YX differs from some common conventions; add a warning so readers can track the signs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the overlap formulas are explicit computations backed by independent external results.

full rationale

The paper's central derivation chain is self-contained in the relevant sense. Theorem 3.7 defines six polynomial eigenbases directly via q-Pochhammer symbols and then computes their overlaps by applying the univariate q-Racah formula (2.27)–(2.28). That univariate formula is imported from Rosengren [R03], an independent external source, not from the paper's own conclusions. The bivariate functions S and T are therefore not fitted inputs disguised as predictions; they are computed products of the univariate overlaps. The TD-pair statement in Proposition 4.3 depends on the irreducibility theorem [IT09, Thm. 1.17(i)] and on a parameter correspondence (4.4) whose detailed derivation is not fully displayed. This is a legitimate correctness risk — if the correspondence or normalization (4.5)–(4.6) is wrong, the structural TD-pair interpretation could fail — but it is not circularity: the cited irreducibility result is external, and the claimed reduction is a computation, not an assumption of the target claim. The self-citations that occur, notably [BB09] for the q-Onsager embedding and [BVZ16] for the split basis, supply explicit, independently checkable constructions; they do not smuggle in the bivariate overlap result, and the embedding is also available from the non-self-cited literature (e.g. [IT09], [Ko12]). Conjecture 4.20 is explicitly labelled a conjecture, so the FL-pair part is not presented as an established derivation. No fitted constants, no definition-in-terms-of-target, and no prediction that is forced by construction were found. The paper's advertised interpretation is genuinely a construction/identification rather than a circular derivation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central construction depends on standard q-Onsager/quantum-loop machinery, the univariate q-Racah overlap formula from Rosengren, and a parameter-transfer assumption from Itô–Terwilliger. No data fitting is involved. The main novel conceptual entity, the factorized B'_2-Leonard pair, is not independently established.

free parameters (6)
  • a
    C* scalar parameterizing alpha = a + a^{-1} in the embedding phi and the q-Racah function labels.
  • b
    C* scalar parameterizing beta = b + b^{-1} in the embedding phi and the q-Racah function labels.
  • u1
    Evaluation parameter of the first tensor factor; labels the representation V_{2j1}(u1).
  • u2
    Evaluation parameter of the second tensor factor; labels the representation V_{2j2}(u2).
  • j1
    Half-integer spin fixing the dimension 2j1 + 1 of the first factor.
  • j2
    Half-integer spin fixing the dimension 2j2 + 1 of the second factor.
assumptions (5)
  • domain assumption q is not a root of unity (Notations, p. 3).
    All representation-theoretic and TD-pair results assume this; degeneracies at roots of unity are excluded.
  • standard math The univariate overlap formula R_k(ell) (Eqs. 2.27-2.28) from [R03] is valid.
    Theorem 3.7 builds bivariate overlaps by applying this formula twice; it is taken as a black box.
  • domain assumption Irreducibility/tensor-product classification of O_q-modules from [IT09, Thm. 1.17(i)] transfers to the present embedding via the parameter map (4.4).
    Prop 4.3's TD-pair characterization rests on this transfer, which is asserted rather than fully proven.
  • standard math Evaluation representations V_{2j}(u) are irreducible for generic u and tensor-product irreducibility follows from [CP91].
    Used to justify basis properties and TD-pair irreducibility in Section 3.3 and Prop 4.3.
  • ad hoc to paper The new B'_2-Leonard pair definitions (Defs 4.17-4.18) are well-posed.
    These definitions are introduced specifically for Conjecture 4.20; no independent existence or classification is provided.
invented entities (1)
  • B'_2-Leonard pair / factorized B'_2-Leonard pair
    purpose: Proposed refined TD-pair structure for u1 = u2; Conjecture 4.20 asserts that (A2, A12; B12, B1) is such a pair.
    The concept is new in this paper and no instance is proven; the only supporting evidence is an explicit conjecture.

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Pith. "Pith review of The quantum loop algebra of $sl_2$ and $q$-Racah type bivariate functions." pith.science (2026). https://pith.science/paper/IYTPBUZN

@misc{pith2026260720043,
  author       = {Pith},
  title        = {Pith review of: The quantum loop algebra of $sl_2$ and $q$-Racah type bivariate functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYTPBUZN}},
  note         = {Machine review of arXiv:2607.20043}
}
abstract

A unified algebraic framework for two different six-parameter families of bivariate $q$-Racah type functions is given using the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$ of $sl_2$. The starting point of the analysis is two left and right coideal subalgebras of $\mathcal{L} U_q sl_2$ and six commutative subalgebras, built from eight elements in $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$. The eight elements depend on two scalars $a,b \in {\mathbb C}^*$, and are diagonalized on a finite-dimensional vector space. The bivariate $q$-Racah type functions are interpreted as the overlap coefficients relating six `distinguished' eigenbases parametrized by $a,b$ of the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$ labeled by the evaluation parameters $u_1,u_2$. Upon certain conditions on $a,b,u_1,u_2$, it is shown that a subset of pairs of elements act as tridiagonal pairs of type I (also called $q$-Racah type). For $u_1/u_2=1$, another subset of pairs of elements are conjectured to act as factorized Leonard pairs. Thus, in both cases corresponding overlap coefficients relating the various eigenbases associated with different pairs are obtained. Some of their properties are also discussed, as well as their relation with known bivariate polynomials of Tratnik type and the rank 2 Askey--Wilson algebra.

Figures

Figures reproduced from arXiv: 2607.20043 by the authors.

Figure 1
Figure 1. • TD pairs of q-Racah type are obtained under certain conditions on a, b, u1, u2; See Propositions 4.3 & 4.5. Moreover, on the line u1/u2 = 1, TD pairs that admit a further refined structure associated with FL pairs of a new type called B′ 2 are identified. In both cases, the overlap coefficients between eigenbases of the TD pairs and FL pairs are explicitly obtained; See Proposition 4.8 and Conjecture 4.20. 1Our st… view at source ↗
Figure 1
Figure 1. Two operators around the same vertex commute. Their associated eigenbases are also indicated at the corresponding vertex. The dashed (resp. doubled) arcs are associated with overlap coefficients given by q-shifted products of q-Racah polynomials. The simple lines are associated with overlap coefficients given by q-Racah polynomials. for vn1,n2 , A2vn1,n2 = (q 2n2−2j2 a + q 2j2−2n2 a −1 )vn1,n2 (3.33) , B1vn1,n2 = (q… view at source ↗

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Works this paper leans on

47 extracted references · 31 linked inside Pith

  1. [1]

    Askey and J

    R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols , SIAM Journal on Mathematical Analysis, 10 (5) (1979) 1008–1016

    1979

  2. [2]

    Baseilhac, Deformed Dolan-Grady relations in quantum integrable models , Nucl

    P. Baseilhac, Deformed Dolan-Grady relations in quantum integrable models , Nucl. Phys. B 709 (2005) 491-521; https://arxiv.org/abs/hep-th/0404149 arXiv:hep-th/0404149

    arXiv 2005

  3. [3]

    P.\ Baseilhac and S.\ Belliard, An attractive basis for the q -Onsager algebra; arXiv:1704.02950 https://arxiv.org/abs/1704.02950

    arXiv

  4. [4]

    P.\ Baseilhac and S.\ Belliard, Generalized q -Onsager algebras and boundary affine Toda field theories , Let.\ Math.\ Phys.\ 93 (2010), 213-228; https://arxiv.org/abs/0906.1215 arXiv:0906.1215

    arXiv 2010

  5. [5]

    De Bie, V

    H. De Bie, V. X. Genest, S. Tsujimoto, L. Vinet, A. Zhedanov, The Bannai-Ito algebra and some applications , J. Phys.: Conf. Ser. 597 012001 (2015) arXiv:1411.3913 https://arxiv.org/abs/1411.3913

    arXiv 2015

  6. [6]

    Baseilhac and X

    P. Baseilhac and X. Martin, A bispectral q -hypergeometric basis for a class of quantum integrable models , Journal of Mathematical Physics 59 011704 (2018), https://arxiv.org/abs/1506.06902 arXiv:1506.06902

    arXiv 2018

  7. [7]

    Baseilhac, L

    P. Baseilhac, L. Vinet and A. Zhedanov, The q -Onsager algebra and multivariable q -special functions, J.Phys.A: Math. Theor. 50 (2017) 395201; https://arxiv.org/abs/1611.0925 arXiv:1611.09250

    arXiv 2017

  8. [8]

    Chari and A

    V. Chari and A. Pressley, Quantum affine algebras , Comm. Math. Phys. 142 (1991), no.\,2, 261--283. 1137064

    1991

Show all 47 references
  1. [9]

    Crampé, L

    N. Crampé, L. Frappat, J. Gaboriaud, L. Poulain d'Andecy, E. Ragoucy and L. Vinet, The Askey--Wilson algebra and its avatars, J. Phys. A: Math. Theor. 54 (2021) 063001; https://arxiv.org/abs/2009.14815 arXiv:2009.14815

    2021 arXiv

  2. [10]

    Crampé, L

    N. Crampé, L. Frappat, J. Gaboriaud and E. Ragoucy, q -deformed Griffiths polynomials of Racah type, J. Phys. A: Math. Theor. 58 (2025) 445205; https://arxiv.org/abs/2407.17016 arXiv:2407.17016

    2025 arXiv

  3. [11]

    Crampé, L

    N. Crampé, L. Frappat, J. Gaboriaud, E. Ragoucy, L. Vinet and M. Zaimi, Griffiths polynomials of Racah type, J. Math. Phys. 65 (2024) 083507; https://arxiv.org/abs/2403.12148 arXiv:2403.12148

    2024 arXiv

  4. [12]

    Crampé, L

    N. Crampé, L. Frappat, L. Poulain d'Andecy and E. Ragoucy, The higher-rank Askey--Wilson algebra and its braid group automorphisms, SIGMA 19 (2023) 077; https://arxiv.org/abs/2303.17677 arXiv:2303.17677

    2023 arXiv

  5. [13]

    Cramp\'e, L

    N. Cramp\'e, L. Frappat and E. Ragoucy, Representations of the rank two Racah algebra and orthogonal multivariate polynomials, Linear Algebra and its Applications 664 (2023) 165–215; https://arxiv.org/abs/2206.01031 arXiv:2206.01031

    2023 arXiv

  6. [14]

    Crampé, J

    N. Crampé, J. Gaboriaud and S. Tsujimoto, Change of basis for the tridiagonal pairs of type II, Nucl. Phys. B 1018 (2025) 117083; https://arxiv.org/abs/22503.01231 arXiv:2503.01231

    2025 arXiv

  7. [15]

    Crampé and M

    N. Crampé and M. Zaimi, Factorized A_2 -Leonard pair , The Ramanujan Journal (2025) 66 25; https://arxiv.org/abs/2312.08312 arXiv:2312.08312

    2025 arXiv

  8. [16]

    De Bie, H

    H. De Bie, H. De Clercq and W. van de Vijver, The higher rank q -deformed Bannai--Ito and Askey--Wilson algebra, Comm. Math. Phys. 374 (2020), 277–316; https://arxiv.org/abs/1805.06642 arXiv:1805.06642

    2020 arXiv

  9. [17]

    Gasper and M

    G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, 2nd edition, 2004

    2004

  10. [18]

    Gasper and M

    G. Gasper and M. Rahman, Some systems of multivariable orthogonal Askey--Wilson polynomials, in: Theory and applications of special functions, pp. 209–219, Dev. Math. 13, Springer, New York, 2005; https://arxiv.org/abs/math/0410249 arXiv:math/0410249

    2005 arXiv

  11. [19]

    Genest, L

    V.X. Genest, L. Vinet and A. Zhedanov, The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states, J. Phys. A: Math. Theor. 46 (2013) 505203; https://arxiv.org/abs/1306.4256 arXiv:1306.4256

    2013 arXiv

  12. [20]

    Griffiths, Orthogonal polynomials on the multinomial distribution, Austral

    R. Griffiths, Orthogonal polynomials on the multinomial distribution, Austral. J. Statist. 13 (1971) 27–35

    1971

  13. [21]

    Groenevelt, Multivariate Askey--Wilson functions and overlap coefficients, Arab

    W. Groenevelt, Multivariate Askey--Wilson functions and overlap coefficients, Arab. J. Math. (2025); arXiv:2504.09735 https://arxiv.org/abs/2504.09735

    2025 arXiv

  14. [22]

    Groenevelt and C

    W. Groenevelt and C. Wagenaar, An Askey--Wilson algebra of rank 2, SIGMA 19 (2023) 008; https://arxiv.org/abs/2206.03986 arXiv:2206.03986

    2023 arXiv

  15. [23]

    Ya. A. Granovskii, and A.S. Zhedanov, Nature of the symmetry group of the 6j --symbol , JETP 67 (1988) 1982--1985

    1988

  16. [24]

    Granovskii and A.S

    Y.I. Granovskii and A.S. Zhedanov, Linear covariance algebra for SL_q(2) , J. Phys. A 26 (1993), L357–L359

    1993

  17. [25]

    Iliev, Bispectral commuting difference operators for multivariable Askey-–Wilson polynomials, Trans

    P. Iliev, Bispectral commuting difference operators for multivariable Askey-–Wilson polynomials, Trans. Amer. Math. Soc. 363 (2011) 1577–1598; https://arxiv.org/abs/0801.4939 arXiv:0801.4939

    2011 arXiv

  18. [26]

    Ito, TD-pairs and the q -Onsager algebra, SUGAKU Volume 65 (2013) Issue 1

    T. Ito, TD-pairs and the q -Onsager algebra, SUGAKU Volume 65 (2013) Issue 1

    2013

  19. [27]

    Ito, The classification of TD-pairs, RIMS Kôkyûroku 1926, 146–164 (2014)

    T. Ito, The classification of TD-pairs, RIMS Kôkyûroku 1926, 146–164 (2014)

    1926

  20. [28]

    Ito and P

    T. Ito and P. Terwilliger, Tridiagonal pairs of q -Racah type , Journal of Algebra 322 (2009) 68-93; https://arxiv.org/abs/0807.0271 arXiv:0807.0271

    2009 arXiv

  21. [30]

    T. Ito, K. Tanabe and P. Terwilliger, Some algebra related to P - and Q-polynomial association schemes , in: Codes and Association Schemes (Piscataway NJ, 1999), Amer. Math. Soc., Providence RI (2001) 167–192. MR1816397

    1999

  22. [31]

    Post and A

    S. Post and A. Walter, A higher rank extension of the Askey--Wilson algebra ; https://arxiv.org/abs/0904.2889 arXiv:1705.01860

    arXiv

  23. [32]

    Koekoek, P

    R. Koekoek, P. A. Lesky and R. F. Swarttouw. Hypergeometric orthogonal polynomials and their q -analogues . Springer-Verlag, Springer Monographs in Mathematics edition, 2010

    2010

  24. [33]

    Koornwinder, Askey–Wilson polynomials as zonal spherical functions on the SU(2) quantum group , SIAM J

    T.H. Koornwinder, Askey–Wilson polynomials as zonal spherical functions on the SU(2) quantum group , SIAM J. Math. Anal. 24 (1993) 795–813

    1993

  25. [34]

    S.\ Kolb, Quantum symmetric Kac–Moody pairs, Adv.\ in Math.\ 267 (2014) 395-469; https://arxiv.org/abs/1207.6036 arXiv:1207.6036

    2014 arXiv

  26. [35]

    H. Rosengren, An elementary approach to 6j -symbols (classical, quantum, rational, trigonometric, and elliptic), The Ramanujan Journal (2007) 13 131--166; arXiv:math/0312310 https://arxiv.org/abs/math/0312310

    2007 arXiv

  27. [36]

    Terwilliger, Leonard pairs and dual polynomial sequences , preprint 1987

    P. Terwilliger, Leonard pairs and dual polynomial sequences , preprint 1987

    1987

  28. [37]

    Editors A.\ N.\ Kirillov, A

    P.\ Terwilliger, Two relations that generalize the q-Serre relations and the Dolan-Grady relations , Proc.\ of the Nagoya 1999 Int.\ workshop on phys.\ and combinatorics. Editors A.\ N.\ Kirillov, A. Tsuchiya, H.\ Umemura. 377-398; https://arxiv.org/abs/math/0307016 arXiv:math/0307016

    1999 arXiv

  29. [38]

    Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl

    P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001) 149--203; arXiv:math/0406555 https://arxiv.org/abs/math/0406555

    2001 arXiv

  30. [39]

    P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition, Journal of Computational and Applied Mathematics 178 (2005) 437--452; https://arxiv.org/abs/math/0306290 arXiv:math/0306290

    2005 arXiv

  31. [40]

    Terwilliger, Leonard pairs from 24 points of view , Rocky Mountain J

    P. Terwilliger, Leonard pairs from 24 points of view , Rocky Mountain J. Math. 32 (2002) 827-888, arXiv:math/0406577 https://arxiv.org/abs/math/0406577

    2002 arXiv

  32. [41]

    Terwilliger, The Universal Askey–Wilson Algebra , SIGMA 7 (2011) 069; https://arxiv.org/abs/1104.2813 arXiv:1104.2813

    P. Terwilliger, The Universal Askey–Wilson Algebra , SIGMA 7 (2011) 069; https://arxiv.org/abs/1104.2813 arXiv:1104.2813

    2011 arXiv

  33. [42]

    Terwilliger, The q -Onsager algebra and its alternating central extension , Nucl

    P. Terwilliger, The q -Onsager algebra and its alternating central extension , Nucl. Phys. B 975 (2022) 115662; arXiv:2106.14041 https://arxiv.org/abs/2106.14041

    2022 arXiv

  34. [43]

    Terwilliger and P

    P. Terwilliger and P. Iliev, The Rahman polynomials and the Lie algebra sl_3( C) , Trans. Amer. Math. Soc. 364 (2012), no. 8, 4225--4238; arxiv.1006.5062 https://arxiv.org/abs/1006.5062

    2012 arXiv

  35. [44]

    Terwilliger and R

    P. Terwilliger and R. Vidunas, Leonard pairs and the Askey-Wilson relations , J. Algebra and Appl. 03 (2004) 411-426, arXiv:math/0305356 https://arxiv.org/abs/math/0305356

    2004 arXiv

  36. [45]

    Tratnik, Some multivariable orthogonal polynomials of the Askey tableau continuous families, J

    M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau continuous families, J. Math. Phys. 32 (1991) 2065–2073

    1991

  37. [46]

    M. V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau discrete families, J. Math. Phys. 32 (1991) 2337

    1991

  38. [47]

    Wiegmann and A.V

    P.B. Wiegmann and A.V. Zabrodin, Algebraization of difference eigenvalue equations related to U_q(sl_2) , Nucl. Phys. B 451 (1995) 699; https://arxiv.org/abs/cond-mat/9501129 arXiv:cond-mat/9501129

    1995 arXiv

  39. [48]

    Zhedanov, ``Hidden symmetry'' of the Askey-Wilson polynomials , Theoret

    A.S. Zhedanov, ``Hidden symmetry'' of the Askey-Wilson polynomials , Theoret. and Math. Phys. 89 (1991) 1146-1157

    1991

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