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Type II Multi-indexed Little q-Jacobi and Little q-Laguerre Polynomials [ src, pdf ]

arXiv:2402.17272v1 [math-ph]
S.Odake
23 pages, DPSU-24-1

Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme [ src, ps, pdf ] [v1: pdf ]

arXiv:2301.00678v2 [math.CA]
S.Odake
28 pages, DPSU-22-3
typo (2301.00678v1, 44 pages):
* page 41, sec.A.4, 2nd line : discrete Hermite ==> discrete q-Hermite (two places)

New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations [ src, ps, pdf ]

arXiv:2209.12353v2 [math-ph]
S.Odake
50 pages, DPSU-22-2
Prog. Theor. Exp. Phys. 2023 (2023) 073A01 (39pp)
typo (2209.12353v2):
* page 18, (4.15), 3rd line, end: ), ==> ) (deleta ",")
* page 21, (4.33), 3rd line: \check{\Xi}^{Q,monic} ==> \check{\Xi}^{Q monic} (delete ",")
* page 27, A.1.2, 2nd line, end : \rho=1 ==> \rho=1, (add ",")

``Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme [ src, ps, pdf ]

arXiv:2207.14479v1 [math.CA]
S.Odake and R.Sasaki
31 pages, DPSU-22-1

Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts [v2: src, ps, pdf ] [v1(typo corrected): src, pdf ]

arXiv:2112.09358v2 [math-ph]
S.Odake
32 pages, DPSU-21-2
J. Math. Phys. 64 (2023) 053503 (21pp)
typo (2112.09358v2):
* page 4, 1st line above (2.1) : P_n(y)'s ==> P_n(\eta)'s
* page 8, 4th line : following sections ==> next section
* page 11, 6th line from bottom : d^{II}_{s_{II}} ==> d^{II}_{M_{II}}
* page 14, Remark 3.1, 2nd line : are ==> is
* page 32, [26], 2nd line : arXiv:arXiv ==> arXiv

Markov Chains Generated by Convolutions of Orthogonality Measures [ src, ps, pdf ]

arXiv:2106.04082v3 [math.PR]
S.Odake and R.Sasaki
46 pages, DPSU-21-1
J. Phys. A: Math. Theor. 55 (2022) 275201 (42pp)
((4.76), next eq. of (4.79), (5.6), (5.31) : typos by publisher)

Free Oscillator Realization of the Laguerre Polynomial [ src, ps, pdf ]

arXiv:2008.10756v2 [math-ph]
S.Odake
8 pages, DPSU-20-2
Theoret. and Math. Phys. 210 (2022) 1-7

Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems [ src, ps, pdf ]

arXiv:2003.00219v2 [math-ph]
S.Odake
25 pages, DPSU-20-1
J. Phys. A: Math. Theor. 53 (2020) 365202 (21pp)

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types [ src, ps, pdf ]

arXiv:1912.12381v1 [math-ph]
S.Odake
40 pages, DPSU-19-2
J. Math. Phys. 61 (2020) 053505 (25pp)

Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types [ src, ps, pdf ]

arXiv:1907.12218v2 [math-ph]
S.Odake
27 pages, DPSU-19-1
Prog. Theor. Exp. Phys. 2019 (2019) 123A01 (20pp)
typo (1907.12218v2):
* page 1, abstract, line 5 : are ==> is
* page 1, section 1, line 4 : harmonica ==> harmonic
* page 3, line 7 from bottom : system is ==> system are
* page 8, eq.(3.6), 1 line : type II ==> type II : (add a colon)
* page 17, eq.(4.12), LHS of 2nd eq. : r^{I}_j ==> r_j (delete a superscript I)

Dual Polynomials of the Multi-Indexed (q-)Racah Orthogonal Polynomials [ src, ps, pdf ]

arXiv:1805.00345v2 [math-ph]
S.Odake
31 pages, DPSU-18-2
Prog. Theor. Exp. Phys. 2018 (2018) 073A02 (23pp)

typo (1805.00345v2):
* page 22, line 11 : difference equations ==> differential or difference equations
* page 27, ref.[3] : 705718 ==> 705-718
* page 31, ref.[48] : 1606.0283 ==> 1606.02836

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and q-Racah types [ src, ps, pdf ]

arXiv:1804.10352v2 [math-ph]
S.Odake
45 pages, DPSU-18-1
J. Math. Phys. 60 (2019) 023508 (30pp)

Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts [ src, ps, pdf ]

arXiv:1708.01830v2 [math-ph]
S.Odake
37 pages, DPSU-17-2
Prog. Theor. Exp. Phys. 2017(12) (2017) 123A02 (30pp)

typo (1708.01830v2):
* page 13, eq.(3.42) : (qd^{-\frac12})^{\text{v}} ==> (qd^{-\frac12})^{-\text{v}}
* page 14, 1 line above (3.45) : is ==> are

New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics [ src, ps, pdf ]

arXiv:1702.03078v2 [math-ph]
S.Odake
43 pages, DPSU-17-1
Prog. Theor. Exp. Phys. 2017(5) (2017) 053A01 (36pp)

typo (1702.03078v2):
* page 12, eq.(2.64) : W ==> W_C

Simplified Expressions of the Multi-indexed Laguerre and Jacobi Polynomials [v3: src, pdf ] [v2: src, ps, pdf ]

arXiv:1612.00927v3 [math.CA] (v3:SIGMA version, v2: article.cls)
S.Odake and R.Sasaki
10 pages (14 pages: v2), DPSU-16-4
SIGMA 13 (2017) 020 (10pp)

Multi-indexed Meixner and Little q-Jacobi (Laguerre) Polynomials [ src, ps, pdf ]

arXiv:1610.09854v2 [math.CA]
S.Odake and R.Sasaki
29 pages, DPSU-16-3
J. Phys. A: Math. Theor. 50 (2017) 165204 (23pp).

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators, [ src, ps, pdf ]

arXiv:1606.02836v1 [math-ph]
S.Odake
33 pages, DPSU-16-2
J. Math. Phys. 57 (2016) 113503 (22pp)

typo (1606.02836v1):
* page 9, eq.(3.21) 2nd line : p_{Kj})) ==> p_{Kj})

Orthogonal Polynomials from Hermitian Matrices II [ src, ps, pdf ]

arXiv:1604.00714v2 [math.CA]
S.Odake and R.Sasaki
65 pages, DPSU-16-1
J. Math. Phys. 59 (2018) 013504 (42pp)

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III [ src, ps, pdf ]

arXiv:1509.08213v2 [math-ph]
S.Odake
37 pages, DPSU-15-1
J. Math. Phys. 57 (2016) 023514 (24pp).

typo (1509.08213v2):
* page 2, 3 lines above eq.(1.1) : multi-index ==> multi-indexed
* page 6, eq.(2.13) : \mathbb{Z}_{n\geq 0} ==> \mathbb{Z_{\geq 0}
* page 9, eq.(3.3) : a^{i,j}_r ==> a^{ij}_r (two places)
* page 13, eq.(3.24), 2nd equation : \Gamma_j ==> \Gamma
* page 29, eq.(B.2), \mathcal{H} in the 1st line : sin x^2 ==> sin^2 x, cos x^2 ==> cos^2 x

Reflectionless Potentials for Difference Schr\"odinger Equations [ src, ps, pdf ]

arXiv:1411.2307v2 [math-ph]
S.Odake and R.Sasaki
24 pages, DPSU-14-2
J. Phys. A: Math. Theor. 48 (2015) 115204 (21pp).

Comment: Eq.(3.54) should be modified and/or some condition should be imposed.

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II [ src, ps, pdf, data ]

arXiv:1410.8236v2 [math-ph]
S.Odake
27 pages, DPSU-14-3
J. Math. Phys. 56 (2015) 053506 (18pp).

typo (1410.8236v2):
* page 6, eq.(2.13) : \delta_{m n+k}, ==> \delta_{m,n+k}
* page 7, 1 line above eq.(3.1) : g(z) ==> g(x)
* page 9, 2nd line : Remark 5 ==> Remark 6
* page 9, Remark 4, 3rd line : Y(\eta)=2\Xi_{\ell}(\eta) ==> Y(\eta)=2\partial_{\eta}\Xi_{\ell}(\eta)
* page 11, 2nd line from bottom : g(z) ==> g(x)
* page 23, eq.(B.12) first line, denominator in r.h.s : (1+q)\sigma_1 ==> (1+q)\sigma_2
* page 23, eq.(B.12) last line : A, ==> A.

Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm [ src, ps, pdf ]

arXiv:1406.2768v2 [math-ph]
S.Odake and R.Sasaki
37 pages, DPSU-14-1
J. Math. Phys. 56 (2015) 073502 (25pp)

Equivalences of the Multi-Indexed Orthogonal Polynomials [ src, ps, pdf ]

arXiv:1309.2346v2 [math-ph]
S.Odake
25 pages, DPSU-13-4
J. Math. Phys. 55 (2014) 013502 (17pp).

typo (1309.2346v2):
* page 20, eq.(A.39) : P_{\mathcal{D}} ==> P_{\mathcal{D},n}
* page 21, 1 line above eq.(A.42) : c^{I}_n, c^{II}_n ==> c^{I}_{\text{v}}, c^{II}_{\text{v}}

Casoratian Identities for the Wilson and Askey-Wilson Polynomials [ src, ps, pdf ]

arXiv:1308.4240v2 [math-ph]
S.Odake and R.Sasaki
31 pages, 2 figures, DPSU-13-2
J. Approx. Theory 193 (2015) 184-209.

Non-polynomial extensions of solvable potentials a la Abraham-Moses [ src, ps, pdf ]

arXiv:1307.0910v1 [math-ph]
S.Odake and R.Sasaki
29 pages, DPSU-13-3
J. Math. Phys. 54 (2013) 102106 (19pp).

typo (1307.0910v1):
* page 13, eq.(2.56) : \varphi_1 ==> \phi_d (2 places)

Recurrence Relations of the Multi-Indexed Orthogonal Polynomials [ src, ps, pdf ]

arXiv:1303.5820v2 [math-ph]
S.Odake
27 pages, DPSU-13-1
J. Math. Phys. 54 (2013) 083506 (18pp).

typo (1303.5820v2):
* page 11, second step, 4-th line, last : ...(x), ==> ...(x)
* page 15, 7-th line (eq.(3.23), 2nd line from bottom): ...-(\sqrt ... ==> ...-\sqrt ...
* page 19, eq.(A.2) : \tilde{\delta}_{I} ==> \tilde{\delta}^{I}, \tilde{\delta}_{II} ==> \tilde{\delta}^{II}
* page 20, 1 line below eq.(A.8) : \tilde{\delta}_{I,II} ==> \tilde{\delta}^{I,II}

Extensions of solvable potentials with finitely many discrete eigenstates [ src, ps, pdf ]

arXiv:1301.3980v2 [math-ph]
S.Odake and R.Sasaki
22 pages, 3 figure, DPSU-12-4, YITP-12-96
J. Phys. A: Math. Theor. 46 (2013) 235205 (15pp).

Krein-Adler transformations for shape-invariant potentials and pseudo virtual states [ src, ps, pdf ]

arXiv:1212.6595v2 [math-ph]
S.Odake and R.Sasaki
33 pages, 1 figure, DPSU-12-3, YITP-12-85
J. Phys. A: Math. Theor. 46 (2013) 245201 (24pp).

Multi-indexed Wilson and Askey-Wilson Polynomials [ src, ps, pdf ]

arXiv:1207.5584v2 [math-ph]
S.Odake and R.Sasaki
30 pages, DPSU-12-2, YITP-12-48
J. Phys. A: Math. Theor. 46 (2013) 045204 (22pp).

typo (1207.5584v2):
* (2.12) l.h.s : ...,gf_n] ==> ...,gf_n](x)
* (3.46) 1st line : P_{D,n}P_{D,n} ==> P_{D,n}P_{D,m}
* (A.5) 2nd line : P_{D} ==> P_{D,n}

Multi-indexed (q-)Racah Polynomials [ src, ps, pdf ]

arXiv:1203.5868v2 [math-ph]
S.Odake and R.Sasaki
29 pages, DPSU-12-1, YITP-12-18
J. Phys. A: Math. Theor. 45 (2012) 385201 (21pp).
(arXiv:1203.5868v1 [math-ph] 39 pages [pdf, comment])

typo :
* 1203.5868v2 (page 13, line 8 from the bottom), 1203.5868v1(page 10, line 5) :
\hat{A}_{d_1}^{\dagger}\hat{A}_{d_1} ==> \hat{\mathcal{A}}_{d_1}^{\dagger}\hat{\mathcal{A}}_{d_1}

Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials [ src, ps, pdf ]

arXiv:1105.0508v2 [math-ph]
S.Odake and R.Sasaki
7 pages, 1 figure, DPSU-11-4, YITP-11-52.
Phys. Lett. B702 (2011) 164-170.

typo (1105.0508v2):
* 3rd paragraph in sec.4, line 3 : [4,20,7,8] ==> [4,20,7,8,34,35]
* page 2, right column, line 12: \log|(W ==> \log|W

Discrete Quantum Mechanics [ src, ps, pdf ]

arXiv:1104.0473v2 [math-ph]
S.Odake and R.Sasaki
61 pages, 1 figure, DPSU-11-3, YITP-11-35.
J. Phys. A: Math. Theor. 44 (2011) 353001 (47pp)

typo (1104.0473v2):
* eq.(2.132), c_1(\eta,\lambda) for H : -\frac{1}{2} ==> -\frac{\eta}{2}

The Exceptional (X_{\ell}) (q)-Racah Polynomials [ src, ps, pdf ]

arXiv:1102.0812v1 [math-ph]
S.Odake and R.Sasaki
25 pages, DPSU-11-2, YITP-11-18.
Prog. Theor. Phys. 125 (2011) 851-870.

typo (1102.0812v1):
* eq.(6.12), 2nd line : dHdqH ==> dH,dqH
* 1 line above eq.(5.15) : are ==> is

Dual Christoffel transformations [ src, ps, pdf ]

arXiv:1101.5468v1 [math-ph] (revised version tex src only, pdf )
S.Odake and R.Sasaki
40 pages, 4 figures, DPSU-11-1, YITP-11-7.
Prog. Theor. Phys. 126 (2011) 1-34.

typo (1101.5468v1):
* page 31, 2 line below eq.(A.33) : x=0,1,...,x_{max}^{\ell} ==> x=0,1,...,x_{max}^{\ell}+1,
* page 31, 2 line below eq.(A.33) : 0\leq x\leq x_{max} ==> 0\leq x\leq x_{max}^{\ell}+1

A new family of shape invariantly deformed Darboux-P\"oschl-Teller potentials with continuous \ell [ src, ps, pdf ]

arXiv:1007.3800v2 [math-ph]
S.Odake and R.Sasaki
19 pages, DPSU-10-3, YITP-10-64.
J. Phys. A44 (2011) 195203 (14pp)

Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations [ src, ps, pdf ]

arXiv:1004.0544v2 [math-ph]
S.Odake and R.Sasaki
24 pages, DPSU-10-2, YITP-10-19.
J. Phys. A 43 (2010) 335201 (18pp).

typo (1004.0544v2):
* page 12, 1 line below eq.(3.1) : their the Schrodinger ==> their Schodinger
* page 20, 10-th line : from a known ones ==> from known ones
* page 23, ref.[30] : acad. ==> Acad.
* page 23, ref.[32] : Pramana ==> Pramana J. Phys.
remark:
* eqs.(2.73) and (2.74): It is assumed that 3F2 and 4F3 are finite sums.

Modification of Crum's Theorem for `Discrete' Quantum Mechanics [ src, ps, pdf ]

arXiv:1004.0289v2 [math-ph]
Leonor Garcia-Gutierrez, S.Odake and R.Sasaki
31 pages, 2 figures, DPSU-10-1, YITP-10-15.
Prog. Theor. Phys. 124 (2010) 1-26.

typo (1004.0289v2):
* page 2, 6-th line from below : algebra of these algebras ==> algebra of these systems
* page 16, 9-th line : deleted level ==> deleted levels
* eq.(3.48) : V(x)^* ==> V^*(x) (2 places)

Properties of the exceptional (X_{\ell}) Laguerre and Jacobi polynomials [ src, ps, pdf ]

arXiv:0912.5447v4 [math-ph]
C.-L.Ho, S.Odake and R.Sasaki
30 pages, DPSU-09-7, YITP-09-70.
SIGMA 7 (2011) 107 (24pp).

Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials [ src, ps, pdf ]

arXiv:0911.3442v1 [math-ph]
S.Odake and R.Sasaki
4 pages, DPSU-09-6, YITP-09-74,
Phys. Lett. B684 (2010) 173-176.

typo (arXiv:0911.3442v1):
* end of sec.2 : see a recent paper [12] ==> see a recent paper [6]

Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials [ src, ps, pdf ]

arXiv:0911.1585v1 [math-ph]
S.Odake and R.Sasaki
13 pages, DPSU-09-5, YITP-09-69.
J. Math. Phys. 51 (2010) 053513 (9pp)

Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials [ src, ps, pdf ]

arXiv:0909.3668v2 [math-ph]
S.Odake and R.Sasaki
7 pages, DPSU-09-4, YITP-09-52.
Phys. Lett. B682 (2009) 130-136.

Infinitely many shape invariant potentials and new orthogonal polynomials [ src, ps, pdf ]

arXiv:0906.0142v2 [math-ph]
S.Odake and R.Sasaki
4 pages, DPSU-09-3, YITP-09-36.
Phys. Lett. B679 (2009) 414-417.

Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism [ src, ps, pdf ]

arXiv:0903.2604v1 [math-ph]
S.Odake and R.Sasaki
LaTeX2e 32pages, DPSU-09-2, YITP-09-14.
J. Math. Phys 51 (2010) 083502 (24pp).

Crum's Theorem for `Discrete' Quantum Mechanics [ src, ps, pdf ]

arXiv:0902.2593v2 [math-ph]
S.Odake and R.Sasaki
LaTeX2e 13pages, DPSU-09-1, YITP-09-12.
Prog. Theor. Phys 122 (2009) 1067-1079.

Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states [ src, ps, pdf ]

arXiv:0802.1075v1 [quant-ph]
S.Odake and R.Sasaki
LaTeX2e 46pages, 2figures, DPSU-08-1, YITP-08-1.
Prog. Theor. Phys. 119 (2008) 663-700

Orthogonal Polynomials from Hermitian Matrices [ src, ps, pdf ]

arXiv:0712.4106v2 [math.CA]
S.Odake and R.Sasaki
LaTeX2e 53pages, DPSU-07-5, YITP-07-91.
J. Math. Phys. 49 (2008) 053503 (43pp)

typo (arXiv:0712.4106v2):
* eq.(5.55), LHS : \phi_0(x;\lambda) ==> \phi_0(x;\lambda)^2
* eq.(5.73), argument of R_n : \epsilon' (1+d+\eta(x;\lambda)) ==> 1+d+\epsilon' \eta(x;\lambda)
* eq.(5.73), last part : ...,dc^{-1}) ==> ...,dc^{-1}|q)
* eq.(5.80), last part : ...q^{-1}))(1+\tilde{d})) ==> ...q^{-1})(1+\tilde{d})))
* eq.(5.122) : B(...)=q^{-1} B(...), D(...)=q D(...) ==> B(...)=q^{-2} B(...), D(...)=D(...)
* eq.(5.210), 2nd eq. : d_n(\lambda)^2=... ==> d_n(\lambda)^2=q^n ...
* page 48, 2 lines below eq.(A.9) : the the ==> the
* page 49, 2nd line : the the ==> the
* page 53, ref.23, last line : 319324 ==> 319-324
* The dual q-Meixner polynomial in sec.5.2.4 and dual q-Charlier polynomial in sec.5.2.8 should be deleted because the hermiticity of the Hamiltonian is lost for these two cases. ==> See arXiv:1604.00714.

q-oscillator from the q-Hermite Polynomial [ src, ps, pdf ]

arXiv:0710.2209v2 [hep-th]
S.Odake and R.Sasaki
12 pages, DPSU-07-4, YITP-07-64.
Phys. Lett. B 663 (2008) 141-145

Multi-Particle Quasi Exactly Solvable Difference Equations [ src, ps, pdf ]

arXiv:0708.0716v1 [nlin.SI]
S.Odake and R.Sasaki
LaTeX2e 12 pages, DPSU-07-3, YITP-07-44.
J. Math. Phys. 48 (2007) 122105 (8pp)

Interpolation of SUSY quantum mechanics [ src, ps, pdf ]

arXiv:0707.0314v1 [math-ph]
S.Odake, Y.Pehlivan and R.Sasaki
LaTeX2e 18 pages, DPSU-07-2, YITP-07-39.
J. Phys. A 40 (2007)11973-11986

Exact Heisenberg operator solutions for multi-particle quantum mechanics [ src, ps, pdf ]

arXiv:0706.0768v1 [quant-ph]
S.Odake and R.Sasaki
LaTeX2e 17 pages, DPSU-07-1, YITP-07-26.
J. Math. Phys. 48 (2007) 082106 (12pp)

typo (arXiv:0706.0768v1):
* recursion ==> recurrence
* 2 lines above eq.(3.7) : an ==> any
* 1 line below eq.(3.33) : the the ==> the

Exact solution in the Heisenberg picture and annihilation-creation operators [ src, ps, pdf ]

quant-ph/0605221v2
S.Odake and R.Sasaki
LaTeX2e 10 pages, DPSU-06-2, YITP-06-24.
Phys. Lett. B 641 (2006) 112-117.

Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics [ src, ps, pdf ]

quant-ph/0605215v1
S.Odake and R.Sasaki
LaTeX2e 43 pages, DPSU-06-1, YITP-06-23.
J. Math. Phys. 47 (2006) 102102 (33pp)

typo (quant-ph/0605215v1):
* eq.(2.49) : C_n ==> C_k
* eq.(3.100), RHS : -2(\mathcal{H}_c^2+...) ==> -(2\mathcal{H}_c^2+...)
* eq.(3.105), RHS : -2(\mathcal{H}^2+...) ==> -(2\mathcal{H}^2+...)
* 1 line above eq.(C.14) : Askey-Wilson Hahn ==> Askey-Wilson

Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials [ src, ps, pdf ]

hep-th/0512155v1
S.Odake and R.Sasaki
LaTeX2e 16 pages, 1 figure, DPSU-05-2.
Prog. Theor. Phys. 114 (2005) 1245-1260.

Equilibrium Positions and Eigenfunctions of Shape Invariant (`Discrete') Quantum Mechanics [ src, ps, pdf ]

hep-th/0505070v1
S.Odake and R.Sasaki
LaTeX2e 30 pages, 1 figure, DPSU-05-1.
Rokko Lectures in Mathematics (Kobe University) 18 (2005) 85-110.

typo (hep-th/0505070v1):
* eq.(109) : g_2 ==> g_2+\tfrac12
* 2 lines below eq.(110) : g_2 ==> g_2+\tfrac12
* 3 lines below eq.(10) : couplings ==> coupling
* 1 line below eq.(20) : couplings ==> coupling
* section 7, 2nd paragraph, 7-th line : without invariance ==> without shape invariance
* ref.[17] : Aemr ==> Amer

Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials [ src, ps, pdf ]

hep-th/0410109v2
S.Odake and R.Sasaki
LaTeX2e 14 pages, 1 figure, DPSU-04-4, YITP-04-60.
J. Math. Phys. 46 (2005) 063513 (10pp).

Shape Invariant Potentials in ``Discrete Quantum Mechanics" [ src, ps, pdf ]

hep-th/0410102v1
S.Odake and R.Sasaki
LaTeX2e 15 pages, 1 figure, DPSU-04-3, YITP-04-55.
J. Nonlinear Math.Phys. 12 Supplement 1 (2005) 507-521.

Polynomials Associated with Equilibria of Affine Toda-Sutherland Systems [ src, ps, pdf ]

hep-th/0407259v1
S.Odake and R.Sasaki
LaTeX2e 9 pages, DPSU-04-2, YITP-04-41.
J. Phys. A 37 (2004) 11401-11406.

Equilibria of `Discrete' Integrable Systems and Deformation of Classical Orthogonal Polynomials [ src, ps, pdf ]

hep-th/0407155v2
S.Odake and R.Sasaki
LaTeX2e 45 pages, DPSU-04-1, YITP-04-36.
J. Phys. A 37 (2004) 11841-11876.

typo (0407155v2, J.Phys.A):
* eq.(2.19) (middle part) : \frac{(-x)^j}{r!} ==> \frac{(-x)^j}{j!}
* eq.(4.22) : \varepsilon=\frac{b}{a} ==> \varepsilon=\frac{a}{b}
* 1 line below (3.29) and (4.25) : \frac{\delta}{\varepsilon} ==> \delta\varepsilon
* 3 lines below (2.101) : 2^{-1}\schoose{2n-1}{n}^{-1} ==> \schoose{n-1/2}{n}^{-1}

Polynomials Associated with Equilibrium Positions in Calogero-Moser Systems [ src, ps, pdf ]

hep-th/0206172v1
S.Odake and R.Sasaki
LaTeX2e 41 pages, A Mathematica file ``poly.m" is attached. DPSU-02-1, YITP-02-37.
J. Phys. A 35 (2002) 8283-8314.

Comments on the Deformed W_N Algebra [ src, ps, pdf ]

math.QA/0111230v1
S.Odake
LaTeX2e 10 pages, DPSU-01-2
Int. J. Mod. Phys. B16 (2002) 2055-2064.

On Lepowsky-Wilson's {\mathcal Z}-algebra [ src, ps, pdf ]

math.QA/0005203v1
Y.Hara, M.Jimbo, H.Konno, S.Odake, J.Shiraishi
LaTeX2e 8 pages, DPSU-00-2
Contemporary Mathematics 297 (2002) 143-149.
(Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory edited by S.Berman, P.Fendley, Y-Z.Huang, K.Misra and B.Parshall)

Free Field Construction for the ABF Models in Regime II [ src, ps, pdf ]

math.QA/0001071v2
M.Jimbo, H.Konno, S.Odake, Y.Pugai, J.Shiraishi
LaTeX2e 36 pages, HU-IAS/K-8, DPSU-99-8, RIMS-1266
J. Stat. Phys. 102 (2001) 883-921.

Beyond CFT : Deformed Virasoro and Elliptic Algebras [ src, ps, pdf ]
[ revised points ] v2a[ src, ps, pdf ]

hep-th/9910226v2 (See v2a)
S.Odake
(AMS-)LaTeX(2e) 134 pages, DPSU-99-5
in CRM Series in Mathematical Physics: Theoretical Physics at the End of the Twentieth Century (Lecture Notes of the CRM Summer School, Banff, Alberta) edited by Y. Saint-Aubin and L. Vinet, Springer 2002, 307-449.

typo (9910226v2a):
* BRST Tyupin ==> Tyutin
* 1 line below eq.(2.5) : z^n w^m ==> z^{-n} w^{-m}

Free Field Approach to the Dilute A_L Models [ src, ps, pdf ]

math.QA/9902150v1
Y.Hara, M.Jimbo, H.Konno, S.Odake, J.Shiraishi
(AMS-)LaTeX(2e) 43 pages, HU-IAS/K-7, DPSU-99-2
J. Math. Phys. 40 (1999) 3791-3826.

Elliptic algebra U_{q,p}(\widehat{sl}_2): Drinfeld currents and vertex operators [ src, ps, pdf ]

math.QA/9802002v5
M.Jimbo, H.Konno, S.Odake, J.Shiraishi
(AMS-)LaTeX 49 pages, HU-IAS/K-6, DPSU-98-2
Comm. Math. Phys. 199 (1999) 605-647.

Quasi-Hopf twistors for elliptic quantum groups [ src, ps, pdf ], [ ps(TG) ]

q-alg/9712029v3
M.Jimbo, H.Konno, S.Odake, J.Shiraishi
(AMS-)LaTeX 29 pages, DPSU-97-11
Transformation Groups 4 (1999) 303-327

共形場理論を越えて:変形ビラソロ代数が新しい扉を開く [ ps(not final version) ]

粟田 英資, 久保 晴信, 守田 佳史, 小竹 悟, 白石 潤一
LaTeX ?? pages
日本物理学会誌 Vol.53, No.3 (1998) 170-180.

q-Difference Realization of U_q(sl(M|N)) and Its Application to Free Boson Realization of U_q(\widehat{sl}(2|1)) [ src, ps, pdf ]

q-alg/9701032v1
H.Awata, S.Odake, J.Shiraishi
LaTeX 8 pages, EFI-97-07, DPSU-97-1
Lett. Math. Phys. 42 (1997) 271-279.

Virasoro-type Symmetries in Solvable Models [ src, ps, pdf ]

hep-th/9612233v1
H.Awata, H.Kubo, S.Odake, J.Shiraishi
LaTeX 35 pages, EFI-96-44, DPSU-96-18, UT-764
in Proceedings of the Canada-China Meeting on Theoretical Physics edited by L. Lapointe, M.-L. Ge, Y. Saint-Aubin and L. Vinet, Les Publications CRM, Montreal, 2003, 154-188. [ISBN 2921120372]

Quantum Deformation of the W_N Algebra [ src, ps, pdf ]

q-alg/9612001v1
H.Awata, H.Kubo, S.Odake, J.Shiraishi
LaTeX 18 pages, EFI-96-45, DPSU-96-16, UT-762
in Proceedings of the Canada-China Meeting on Theoretical Physics edited by L. Lapointe, M.-L. Ge, Y. Saint-Aubin and L. Vinet, Les Publications CRM, Montreal, 2003, 112-130. [ISBN 2921120372]

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