Pith. sign in

REVIEW 4 minor 73 references

Type C∨C DAHA and Koornwinder systems mirror type A Macdonald theory for eigenfunctions, but lose the Noumi-Shiraishi series and twisting automorphisms.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-10 22:23 UTC pith:WVXECBUJ

load-bearing objection Solid comparative catalogue of A vs C∨C DAHA/Koornwinder systems that cleanly isolates two real structural gaps; useful reference, not a breakthrough.

arxiv 2607.06738 v1 pith:WVXECBUJ submitted 2026-07-07 hep-th math-phmath.MPmath.QA

Integrable systems inspired by DAHA and DIM algebra: type C^vee C versus type A

L. Bishler , A. Mironov , A. Popolitov This is my paper
classification hep-th math-phmath.MPmath.QA MSC 33D5233D8017B3781R12 PACS 02.30.Ik03.65.Fd
keywords DAHADIM algebraKoornwinder polynomialsMacdonald polynomialsCherednik operatorsvan Diejen-Koornwinder HamiltoniansBaker-Akhiezer functionstype C∨C root systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that the integrable systems built from type C∨C double affine Hecke algebras and their spherical projections stand in the same relation to Koornwinder polynomials that type A DAHA and DIM algebra stand to Macdonald polynomials. Non-symmetric Koornwinder polynomials are the common eigenfunctions of the Cherednik operators; their Weyl-group averages are the ordinary (symmetric) Koornwinder polynomials that diagonalize the van Diejen-Koornwinder Hamiltonians. Almost every structural property that makes the Macdonald theory useful—triangular expansions, recursive construction, orthogonality measures, evaluation formulas, dualities, and a weak form of stability—has a direct counterpart. The two places where the parallel breaks are decisive: there is no factorizing branching rule that would produce a Noumi-Shiraishi-type universal power series, and the DAHA of type C∨C lacks the automorphisms that generate twisted ("integer-ray") systems. The result therefore both enlarges the catalogue of explicitly solvable many-body models and isolates precisely which algebraic features of type A are responsible for the richest part of the Macdonald triad.

Core claim

Non-symmetric and symmetric Koornwinder polynomials are the eigenfunctions of the type-C∨C Cherednik and Koornwinder-van Diejen Hamiltonians respectively, and they possess direct counterparts to the Macdonald triangular expansions, Knop-Sahi recursions, orthogonality measures, evaluation formulas, dualities and weak stability; the only essential failures are the absence of a factorizing branching rule that would yield a Noumi-Shiraishi-type universal series and the absence of enough DAHA automorphisms to produce twisted systems.

What carries the argument

The spherical projection that realises the first Koornwinder Hamiltonian as the Weyl-symmetric combination of Cherednik operators Ci + Ci−1 (eq. 76), together with the recursive action of the affine intertwiners B and Ti that generate all monic non-symmetric Koornwinder polynomials from the constant function.

Load-bearing premise

That the same power-sum construction that turns type-A Cherednik operators into the full tower of Ruijsenaars Hamiltonians continues to produce all higher van Diejen-Koornwinder Hamiltonians from the type-C∨C Cherednik operators, even though the richer automorphism group that generates twisting is missing.

What would settle it

Explicitly compute the second and third commuting van Diejen-Koornwinder operators for n=2 or n=3 and check whether they equal the corresponding power sums of the Cherednik operators restricted to Weyl-symmetric functions; any mismatch would break the claimed parallel.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper systematically compares the integrable systems associated with type-A DAHA/DIM (Cherednik operators, Ruijsenaars–Schneider Hamiltonians, non-symmetric and symmetric Macdonald polynomials, Noumi–Shiraishi series and Baker–Akhiezer functions) with their type-C∨C counterparts (Cherednik operators of type C∨C, Koornwinder–van Diejen Hamiltonians, non-symmetric and symmetric Koornwinder polynomials). It records the parallel structures—triangular expansions, Knop–Sahi-type recursions (92)–(94), orthogonality measures, evaluation formulas, dualities and weak stability—while isolating two genuine structural gaps: the absence of a factorizing branching rule that would produce a Noumi–Shiraishi-type universal series, and the lack of enough DAHA automorphisms to generate twisted systems. Rank-one (Askey–Wilson) specializations and explicit Baker–Akhiezer series for reduced root systems are treated in detail.

Significance. The manuscript supplies a clear, self-contained catalogue of the algebraic properties of non-symmetric and symmetric Koornwinder polynomials that mirrors the well-known Macdonald theory. The recursive constructions, evaluation formulas and dualities are written explicitly and match the literature they cite (Noumi, Sahi, Stokman, Chalykh). By isolating the two places where the C∨C story diverges from type A, the paper clarifies the precise limits of the DIM/spherical-DAHA correspondence beyond type A and provides a useful reference for further work on non-reduced root systems and possible elliptizations.

minor comments (4)
  1. In §2.1.1 the Hecke relation is written (T_i-1)(T_i+t^{-1})=0 while in §3.1.1 it is (T_i-t^2)(T_i+1)=0; a short remark that the two normalizations differ by a rescaling of the generators would help the reader.
  2. Equation (76) presents only the first Koornwinder Hamiltonian as a Weyl-symmetric combination of Cherednik operators; a one-sentence clarification that the higher Hamiltonians (77)–(78) are taken from the classical van Diejen construction (and are not claimed to arise by power sums) would remove any possible ambiguity.
  3. The branching-rule formula (137) cites the very involved coefficients of van Diejen–Emsiz; a pointer to the precise equation number in that reference would make the claim easier to verify.
  4. A few typographical inconsistencies remain (e.g., “eduction” for “reduction” near (57), occasional missing spaces around “=”). A light copy-edit pass would clean them up.

Circularity Check

0 steps flagged

No significant circularity: comparative review of standard DAHA/Koornwinder properties with external citations and self-contained parallels.

full rationale

The paper systematically catalogues eigenfunctions, triangular expansions, Knop–Sahi-type recursions (92)–(94), orthogonality, evaluations, dualities and weak stability for non-symmetric/symmetric Koornwinder polynomials, paralleling the type-A Macdonald case. All load-bearing definitions (DAHA relations (61)–(70), Noumi x-representation (71), Cherednik operators (70), Koornwinder–van Diejen Hamiltonians (74)–(78), Chalykh BA construction (123)–(135)) are taken from the external literature (Noumi, Sahi, Stokman, Chalykh, van Diejen, Koornwinder). Self-citations appear only as type-A templates or earlier triad papers and are never used to justify a C∨C claim. No parameters are fitted to data and recovered as predictions; no uniqueness theorem is imported from the authors’ own prior work; no ansatz is smuggled; nothing is renamed as a new derivation. The two structural gaps (non-factorizing branching, missing automorphisms for twisting) are explicitly isolated as open, not claimed as results. The derivation chain is therefore self-contained against external benchmarks and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The paper works entirely inside the standard axiomatic framework of double affine Hecke algebras and root-system special functions. No free numerical parameters are fitted; the six Koornwinder parameters q,t,a,b,c,d are the usual free parameters of the theory. No new physical entities are postulated. The only background assumptions are the defining relations of DAHA of type C∨C and the existence of the spherical projection that produces the Koornwinder Hamiltonians—both taken from the literature.

free parameters (1)
  • Koornwinder parameters (q,t,a,b,c,d)
    These six complex parameters label the family of systems; they are free by definition of the theory and are never fitted to data.
axioms (3)
  • standard math The defining braid, Hecke and reflection relations of DAHA of type C∨C (eqs. (61)–(69))
    Taken as the starting point of the algebraic construction; standard in the literature since Noumi and Sahi.
  • domain assumption The spherical projection of the Cherednik operators yields the commuting van Diejen-Koornwinder Hamiltonians (eqs. (76)–(78))
    Assumed by direct analogy with the type-A DIM–spherical-DAHA correspondence; the paper notes that the richer automorphism group is missing but still uses the projection for the untwisted case.
  • standard math Chalykh’s periodicity characterisation uniquely determines the Baker-Akhiezer function for any finite root system (including non-reduced BCn)
    Invoked in §5.1; proved in the cited reference [21].

pith-pipeline@v1.1.0-grok45 · 39797 in / 2730 out tokens · 34719 ms · 2026-07-10T22:23:52.633257+00:00 · methodology

0 comments
read the original abstract

The Ding-Iohara-Miki (DIM) algebra (quantum toroidal algebra of $\widehat{gl_1}$) is related to a wide class of quantum many-particle integrable systems, a typical one being the Ruijsenaars trigonometric system with eigenfunctions that are a triad formed by the Noumi-Shiraishi power series, the Macdonald polynomials, and the Baker-Akhiezer multivariable function. Other integrable systems of this type are obtained from the Ruijsenaars system by twisting. At the same time, the Ruijsenaars Hamiltonians are directly related to the Hamiltonians of another quantum integrable system, the Cherednik DAHA Hamiltonians of type $A$ (and their twisted versions in the twisted case), due to the correspondence between the DIM algebra and the spherical DAHA. The eigenfunctions of the DAHA Hamiltonians are non-symmetric Macdonald polynomials. Similarly, there is a class of integrable DAHA Hamiltonians of type $C^\vee C$, the spherical version of which, in turn, allows one to generate integrable Koornwinder Hamiltonians. The eigenfunctions of these two integrable systems are, respectively, non-symmetric and symmetric Koornwinder polynomials, which are our main interest in this paper. Here we consider the cases of both type $A$ and type $C^\vee C$ systems, since they are sufficiently similar, and point out important distinctions between them.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages · 51 internal anchors

  1. [1]

    Commutative families in $W_\infty$, integrable many-body systems and hypergeometric $\tau$-functions

    A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, JHEP,23(2020) 065, arXiv:2306.06623

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  2. [2]

    Pope, L.J

    C.N. Pope, L.J. Romans and X. Shen, Phys.Lett.B236(1989) 173-178; Nucl.Phys.339B(1990) 191-221; Phys.Lett.B242(1990) 401-406; Phys.Lett.B245(1990) 72-78

    work page 1989

  3. [3]

    Awata, M

    H. Awata, M. Fukuma, Y. Matsuo and S. Odake, Prog. Theor. Phys. Suppl.118(1995) 343-374, hep- th/9408158

    work page arXiv 1995

  4. [4]

    Quasifinite highest weight modules over the Lie algebra of differential operators on the circle

    V.G. Kac and A. Radul, Comm.Math.Phys.157(1993) 429-457, hep-th/9308153

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  5. [5]

    Commutative subalgebras from Serre relations

    A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, Phys. Lett.B845(2023) 138122, arXiv:2307.01048

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  6. [6]

    The affine Yangian of $\mathfrak{gl}_1$ revisited

    A. Tsymbaliuk, Adv. Math.304(2017) 583-645, arXiv:1404.5240

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  7. [7]

    W-symmetry, topological vertex and affine Yangian

    T. Proch´ azka, JHEP,10(2016) 077, arXiv:1512.07178 29

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  8. [8]

    Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models

    A. Mironov, A. Morozov, A. Popolitov, JHEP,09(2024) 200, arXiv:2406.16688

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  9. [9]

    J. Ding, K. Iohara, Lett. Math. Phys.41(1997) 181-193, q-alg/9608002

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  10. [10]

    K. Miki, J. Math. Phys.48(2007) 123520

    work page 2007

  11. [11]

    Eisenstein series and quantum affine algebras

    M. Kapranov, Algebraic geometry7, J.Math. Sci.84(1997) 1311-1360, alg-geom/9604018

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  12. [12]

    On the Hall algebra of an elliptic curve, I

    I. Burban, O. Schiffmann, Duke Math. J.161(2012) 1171, arXiv:math/0505148

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  13. [13]

    Drinfeld realization of the elliptic Hall algebra

    O. Schiffmann, J. Algebraic Combin.35(2012) 237-26, arXiv:1004.2575

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

    B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Commun. Math. Phys.356(2017) 285, arXiv:1603.02765

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Ruijsenaars, H

    S.N.M. Ruijsenaars, H. Schneider, Ann.Phys. (NY),170(1986) 370 S.N.M. Ruijsenaars, Comm.Math.Phys.110(1987) 191-213

    work page 1986

  16. [16]

    Miki, Lett

    K. Miki, Lett. Math. Phys.47(1999) 365-378

    work page 1999

  17. [17]

    (t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators

    P. Di Francesco, R. Kedem, Comm. Math. Phys.369(3)(2019) 867-928, arXiv:1704.00154

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  18. [18]

    Cherednik,Double affine Hecke algebras, Vol.319, Cambridge University Press, 2005

    I. Cherednik,Double affine Hecke algebras, Vol.319, Cambridge University Press, 2005

    work page 2005

  19. [19]

    Generating twisted Cherednik eigenfunctions

    A. Mironov, A. Morozov and A. Popolitov, Phys. Lett.B879(2026) 140592, arXiv:2602.21120

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  20. [20]

    A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators

    M. Noumi, J. Shiraishi, arXiv:1206.5364

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    Macdonald polynomials and algebraic integrability

    O. Chalykh, Adv.Math.166(2)(2002) 193-259, math/0212313

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  22. [22]

    Macdonald Duality and the proof of the Quantum Q-system conjecture

    P. Di Francesco, R. Kedem, Selecta Math.30(2024) no.2, 23, arXiv:2112.09798

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  23. [23]

    A basic triad in Macdonald theory

    A. Mironov, A. Morozov, A. Popolitov, Phys. Lett.B869(2025) 139840, arXiv:2411.16517

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  24. [24]

    Orthogonality relations and Cherednik identities for multivariable Baker-Akhiezer functions

    O. Chalykh, P. Etingof, Advances in Mathematics,238(2013) 246-289, arXiv:1111.0515

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  25. [25]

    Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

    O. Chalykh, M. Fairon, J.Geom.Phys.121(2017) 413-437, arXiv:1704.05814

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians

    A. Mironov, A. Morozov, A. Popolitov, Phys. Lett.B863(2025) 139380, arXiv:2410.10685

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  27. [27]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, to appear

    work page

  28. [28]

    Opdam, Acta Mathematica,175(1)(1995) 75–121

    E.M. Opdam, Acta Mathematica,175(1)(1995) 75–121

    work page 1995

  29. [29]

    Macdonald, Asterisque-Societe Mathematique de France,237(1996) 189-208

    I.G. Macdonald, Asterisque-Societe Mathematique de France,237(1996) 189-208

    work page 1996

  30. [30]

    Non-Symmetric Macdonald's Polynomials

    I. Cherednik, IMRN,1995(10)(1995) 483, q-alg/9505029

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  31. [31]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, arXiv:2512.24811

    work page arXiv

  32. [32]

    Twisted Cherednik spectrum as a $q,t$-deformation

    A. Mironov, A. Morozov, A. Popolitov, Nucl. Phys.B1028(2026) 117513, arXiv:2601.10500

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  33. [33]

    Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems

    A. Mironov, A. Morozov, A. Popolitov, Phys. Lett.B877(2026) 140457, arXiv:2601.19878

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  34. [34]

    Macdonald, Invent

    I.G. Macdonald, Invent. Math. 15 (1972) 91–143

    work page 1972

  35. [35]

    Orthogonal polynomials associated with root systems

    I.G. Macdonald, S` eminaire Lotharingien Combin.45(2000), Article B45a, 40 pp, arXiv:math/0011046

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  36. [36]

    Macdonald, SIAM J.Math

    I.G. Macdonald, SIAM J.Math. Anal.13:6(1982) 988-1007

    work page 1982

  37. [37]

    Macdonald's Evaluation Conjectures and Difference Fourier Transform

    I. Cherednik, Inventiones mathematicae,125(1996) 391, q-alg/9412016

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  38. [38]

    Cherednik, The Annals of Mathematics, Second Series,141(1995) 191-216

    I. Cherednik, The Annals of Mathematics, Second Series,141(1995) 191-216

    work page 1995

  39. [39]

    Hypergeometric functions on domains of positivity, Jack polynomials, and applications

    T.H. Koornwinder,Askey-Wilson polynomials for root systems of type BC, in: “Hypergeometric functions on domains of positivity, Jack polynomials, and applications” (Tampa, FL, 1991), Contemp. Math.138 (1992) 189–204

    work page 1991

  40. [40]

    Commuting difference operators with polynomial eigenfunctions

    J.F. van Diejen, Comp. Math.95(1995) 183-233, funct-an/9306002

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  41. [41]

    A combinatorial formula for non-symmetric Macdonald polynomials

    J. Haglund, M. Haiman, N. Loehr, Am.J.Math.130(2)(2008) 359-383, math/0601693 30

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  42. [42]

    F. Knop, S. Sahi, Invent. Math.128(1997) 9-22, q-alg/9610016

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  43. [43]

    A $q$-analogue of the type $A$ Dunkl operator and integral kernel

    T.H. Baker, P.J. Forrester, q-alg/9701039

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [44]

    A reproducing kernel for nonsymmetric Macdonald polynomials

    K. Mimachi, M. Noumi, Duke Math. J.95(1)(1998) 621-634, q-alg/9610014

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  45. [45]

    Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    I.G. Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1995

    work page 1995

  46. [46]

    Intertwining Operators of Double Affine Hecke Algebras

    I. Cherednik, Sel. math., New ser.3(1997) 459–495, q-alg/9605014

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  47. [47]

    Double Affine Hecke Algebras and Difference Fourier Transforms

    I. Cherednik, Invent. Math.152(2003) 213–303, math/0110024

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  48. [48]

    Ruijsenaars, Comm

    S.N. Ruijsenaars, Comm. Math. Phys., 115 (1988) 127-165

    work page 1988

  49. [49]

    Traces of intertwiners for quantum groups and difference equations, I

    P. Etingof, A. Varchenko, Duke Math. J.104(2000) 391-432. , math/9907181 P. Etingof, O. Schiffmann, A. Varchenko, Lett. Math. Phys.62(2002) 143-158 G. Felder, Y. Markov, V. Tarasov, A. Varchenko, Mathematical Physics, Analysis and Geometry,3(2000) 139-177, math/0001184 V. Tarasov, A. Varchenko, IMRN,2000(15)801-829, math/0002132

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  50. [50]

    Duality in elliptic Ruijsenaars system and elliptic symmetric functions

    A. Mironov, A. Morozov, Y. Zenkevich, Eur. Phys. J.C81(2021) 461, arXiv:2103.02508

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  51. [51]

    On the status of DELL systems

    A. Mironov, A. Morozov, Nucl. Phys.B999(2024) 116448, arXiv:2309.06403

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  52. [52]

    Difference Macdonald-Mehta Conjecture

    I. Cherednik, IMRN,1997 (10)(1997) 449-467, q-alg/9702022

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  53. [53]

    On Cherednik-Macdonald-Mehta identities

    P. Etingof, A. Kirillov, Electr.Res.Announc.Amer.Math.Soc.4(1998) 43-47, q-alg/9712051

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  54. [54]

    CMM formula as superintegrability property of unitary model

    A. Mironov, A. Morozov, A. Popolitov, Phys. Rev.D110(2024) 126026, arXiv:2410.03175

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  55. [55]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, Phys. Rev.D113(2026) no.12, 126019, arXiv:2601.17453

    work page arXiv 2026

  56. [56]

    Noumi, Macdonald-Koornwinder polynomials and affine Hecke rings, Surikaisekikenkyusho Kokyuroku 919 (1995), pp

    M. Noumi, Macdonald-Koornwinder polynomials and affine Hecke rings, Surikaisekikenkyusho Kokyuroku 919 (1995), pp. 44–55 (in japanese)

    work page 1995

  57. [57]

    Nonsymmetric Koornwinder polynomials and duality

    S. Sahi, Ann. Math.150(1999) 267–282, q-alg/9710032

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  58. [58]

    Sahi, Some properties of Koornwinder polynomials, Contemp

    S. Sahi, Some properties of Koornwinder polynomials, Contemp. Math. 254 (2000) 395-411

    work page 2000

  59. [59]

    Koornwinder polynomials and affine Hecke algebras

    J. Stokman, IMRN 19 (2000), 1005–1042, math/0002090

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  60. [60]

    Quantum Lax pairs via Dunkl and Cherednik operators

    O. Chalykh, Commun. Math. Phys.369(1)(2019) 261-316, arXiv:1804.01766

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  61. [61]

    $c$-functions and Koornwinder polynomials

    L. Colmenarejo, A. Ram, The Quarterly Journal of Mathematics,76(3)(2025) 983-1031, arXiv:2410.19957

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  62. [62]

    Askey, J

    R. Askey, J. Wilson, Mem. Amer. Math. Soc. vol.319, 1985

    work page 1985

  63. [63]

    Mimachi, Duke Math

    K. Mimachi, Duke Math. J.107(2)(2001) 265–281

    work page 2001

  64. [64]

    Branching Formula for Macdonald-Koornwinder Polynomials

    J.F. van Diejen, E. Emsiz, Journal of Algebra,444(20145) 606-614, arXiv:1408.2280

    work page internal anchor Pith review Pith/arXiv arXiv

  65. [65]

    Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions

    T. Koornwinder, M. Mazzocco, Indagationes Mathematicae (2025), arXiv:2407.17366

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  66. [66]

    The Askey-Wilson function transform

    E. Koelink, J.V. Stokman, IMRN,22(2001) 1203–1227, math/0004053

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  67. [67]

    Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

    J.-E. Bourgine, M. Fukuda, Y. Matsuo, R.-D. Zhu, JHEP, 2017 (2017) 15, arXiv:1709.01954

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  68. [68]

    Deformations of $\mathcal W$ algebras via quantum toroidal algebras

    B. Feigin, M. Jimbo, E. Mukhin, I. Vilkoviskiy, Selecta Mathematica,27(4)(2021) 52, arXiv:2003.04234

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  69. [69]

    Affine Screening Operators, Affine Laumon Spaces, and Conjectures Concerning Non-Stationary Ruijsenaars Functions

    J. Shiraishi, J.Integrable.Syst.4(2019) xyz010, arXiv:1903.07495

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  70. [70]

    Non-Stationary Ruijsenaars Functions for $\kappa=t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra

    M. Fukuda, Y. Ohkubo, J. Shiraishi, SIGMA16(2020) 116, arXiv:2002.00243

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  71. [71]

    Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

    H. Awata, H. Kanno, A. Mironov, A. Morozov, JHEP,08(2020) 150, arXiv:2005.10563

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  72. [72]

    Elliptic triad

    A. Mironov, A. Morozov, A. Popolitov, Z. Zakirova, Phys. Lett. B865(2025) 139467, arXiv:2412.19588

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  73. [73]

    Mironov, A

    A. Mironov, A. Morozov, A. Popolitov, Z. Zakirova, Pisma Zh. Eksp. Teor. Fiz.121(2025) no.9, 788-795, arXiv:2503.07592 31

    work page arXiv 2025