## The history of q-calculus and a new method (2000)

Citations: | 10 - 8 self |

### BibTeX

@TECHREPORT{Ernst00thehistory,

author = {Thomas Ernst},

title = {The history of q-calculus and a new method},

institution = {},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8

### Citations

1665 |
and Complex Analysis
- Rudin, Real
- 1966
(Show Context)
Citation Context ...n[261] (169) Eq(z) = ∞� k=0 The q-difference equation for Eq(z) is 1 {k}q! zk . (170) DqEq(az) = aEq(az). If |q| > 1, Eq(z) is an entire function, which by the Weierstrass factorization theorem [12], =-=[761]-=- can be expressed as the infinite product (171) Eq(z) = e g(z) ∞� Epn( z ), zn n=0 where {zn} ∞ n=0 are the zeros of Eq(z), listed according to their multiplicities, {pn} ∞ n=0 is a sequence of nonneg... |

1540 |
Exactly solved models in statistical mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...ISTORY OF q-CALCULUS AND A NEW METHOD. 103 Remark 25. For a connection between the Rogers-Ramanujan identities, statistical mechanics, thermodynamics and elliptic functions, the reader is referred to =-=[88]-=-. The physicist Baxter independently derived the Rogers-Ramanujan identities in 1979 when working with the socalled hard hexagon model. The Rogers-Ramanujan identities are also useful in finite type i... |

890 |
Symmetric functions and Hall polynomials
- Macdonald
- 1995
(Show Context)
Citation Context ...ons, generalized Vandermonde determinants and representation theory. The theory of symmetric functions appeared first in Newtons Arithmetica Universalis. A modern account is given by Macdonald (1995) =-=[594]-=- which we shall use as reference for matters of notation. The theory of symmetric functions is intimately connected to the symmetric group as was outlined in [594]. The basic theory of the symmetric g... |

850 |
Quantum groups
- Drinfeld
- 1987
(Show Context)
Citation Context ...y of the Pauli principle in doubt. Finally this problem was solved by letting every quark exist in three different disguises or colours. Quantum groups were introduced in the mid-eighties by Drinfeld =-=[243]-=-, Jimbo [475] 1985, [476] 1986 (connection between trigonometric solutions of the Yang-Baxter equation and the Hecke algebra and the corresponding symmetric representation of the q-analogue of gℓN+1),... |

705 |
Basic Hypergeometric Series
- Gasper, Rahman
- 2004
(Show Context)
Citation Context ... a new method for q-hypergeometric series. Of course it is not possible to cover all the details in this vast subject in only one book and the reader is invited to study the book by Gasper and Rahman =-=[343]-=- for additional information. Some parts of chapter 2 are based on the first chapters of this book. In section 2.25 a new q-Taylor formula is presented. Furthermore generalized Vandermonde determinants... |

621 | Special functions - Andrews, Askey, et al. - 1999 |

485 | The representation theory of the symmetric group - James, Kerber - 1981 |

406 |
Quantum Groups
- Kassel
- 1995
(Show Context)
Citation Context ...e q-binomial coefficients. (248) D k q x − qk xD k q (249) D k q x − xDk q k−1 = {k}qDq . k−1 = T {k}qDq . We now prove a variation of the q-binomial theorem, which is applicable to the quantum plane =-=[507]-=-. Theorem 2.20. Let A and B be linear operators on P with (250) BA = qAB. Then (251) (A + B) n = n� k=0 � � n A k q k B n−k , n = 0, 1, 2, . . . . Proof. The theorem is obviously true for n = 1. Assum... |

385 |
Confluent hypergeometric functions
- Slater
- 1960
(Show Context)
Citation Context ... . . . , br n!(b1)n . . . (br)n n . Denote the series (59) �∞ n=0 Anzn . The quotient An+1 is a rational funcAn tion R(n) (Askey). The following notation for the gamma function will sometimes be used =-=[810]-=-: � � a1, . . . , ap (60) Γ b1, . . . , br n=0 ≡ Γ(a1) . . . Γ(ap) Γ(b1) . . . Γ(br) . An r+1Fr series is called k- balanced if b1 + . . . + br = k + a1 + . . . + ar+1 and z = 1; and a 1-balanced seri... |

344 |
An Introduction to Combinatorial Analysis
- Riordan
- 1980
(Show Context)
Citation Context ...nce products of Pochhammer symbols occur so often, we shall frequently use the more compact notation m� (47) (a1, a2, . . . , am)n ≡ (aj)n. We will take some of the following definitions from Riordan =-=[748]-=-. James Stirling (1692-1770) was a disciple of Newton [246, p. 17] who invented the Stirling numbers of the first and second kind s(n, k), S(n, k), which are defined as follows. s(0, 0) = S(0, 0) = 1 ... |

272 |
Group Theory and This Applications to Physical Problem
- Hamermesh
- 1962
(Show Context)
Citation Context ...imensions of the representations associated with conjugate partitions are equal. There are other methods to compute the irreducible characters χλ(µ). One method uses the so-called lattice permutation =-=[412]-=-, [750]. A graphical construction is given in [802]. Let Mλ be the permutation module affording Ψλ. Mλ has a natural basis {fP| ¯ P = λ} permuted by S m . Since χλ has multiplicity 1 in Ψλ, there is a... |

226 |
A.B.,“Current Algebra and Wess-Zumino Model in Two Dimensions
- Knizhnik, Zamolodchikov
- 1984
(Show Context)
Citation Context ...urvey of summation formulae for basic hypergeometric series. In 1975 Dashen and Frishman [221] found an important special case of the Knizhnik-Zamolodchikov (KZ) equation, which was rediscovered 1984 =-=[526]-=- in the general framework of the WZNW model [62]. In 1992 I.Frenkel and Reshetikin [325] started to study quantum affine algebras and holonomic difference equations, in particular the KZ equation in c... |

224 |
Integrals of nonlinear equations of evolution and solitary waves
- Lax
- 1968
(Show Context)
Citation Context ...tion was generated by computer calculations. This story was captivatingly narrated by one of the authors Kruskal M [550]. The relation of non-linear equations to linear ones was explained best by Lax =-=[570]-=-. More general hierarchies are obtained for operators of arbitrary orders and are called generalized KdV hierarchies. The KP equation (5), which has the Weierstrass elliptic function as solution, is a... |

222 | An Introduction to the Theory of Numbers - Niven, Zuckerman, et al. - 2000 |

213 |
Lecons sur la théorie générale des surfaces
- Darboux
- 1894
(Show Context)
Citation Context ...-algebra approach, due to Kalnins, Miller and coworkers, and Floreanini and Vinet uses only the q-analogue of the Lie algebra [531, p. 95]. The factorization method of Darboux-Schrödinger-Infeld-Hull =-=[219]-=-, [453], [500] makes it possible to compute the eigenvalues and eigenfunctions for families of Schrödinger operators. The theory of commutative ordinary differential operators, which predates the work... |

183 | An algorithmic proof theory for hypergeometric (ordinary and q) multisum/integral identities
- Wilf, Zeilberger
- 1992
(Show Context)
Citation Context ...ned its workings to a broad audience in her paper [269]. In 1991 Zeilberger [958] found a method that did the same job as sister Celine’s algorithm, but a great deal faster. In 1992 Wilf & Zeilberger =-=[937]-=- showed that every q-identity, with a fixed number qsTHE HISTORY OF q-CALCULUS AND A NEW METHOD. 109 of summations and/or integration signs, possesses a short, computerconstructible proof, and gave a ... |

175 |
Stable Bundles and Integrable Systems
- Hitchin
- 1987
(Show Context)
Citation Context ...487]. (57) Kac-van Moerbeke (KM) system [281], [355]. (58) Lotka-Volterra system [281]. (59) Regge-Wheeler equation [63]. (60) Darboux-Zakharov-Manakov (DZM) system [220], [111]. (61) Hitchin systems =-=[435]-=-, [508], [665], [854]. (62) Gaudin model [345], [854], [794]. (63) Haldane-Shastry spin chain [82]. (64) Garnier system [338], [728]. (65) Holt system [878]. (66) Kowalewski top [409], [746], [713], [... |

163 |
Girvin (Eds.), The Quantum Hall Effect
- Prange, M
- 1990
(Show Context)
Citation Context ...nants. 4.1. Applications of Schur functions and Schur polynomials. The Schur functions are particularly relevant to discussions of the quantum Hall effect [845], [785]. The quantum Hall effect [845], =-=[717]-=- was discovered on about the hundredth anniversary of Hall’s original work in 1980 by von Klitzing, Dorda and Pepper. The quantum Hall effect is a two-dimensional electron gas in a strong magnetic fie... |

158 |
The umbral calculus
- Roman, Rota
- 1978
(Show Context)
Citation Context ...of Hermite polynomials. In 1971 Andrews [25] introduced the concept of an Eulerian family of polynomials. A few years after the umbral calculus was developed by G.C. Rota and his collaborators [757], =-=[752]-=-, the q-umbral calculus was explored in [203], [205], [447], [753], [754]. In 1982 [753] a q-Euler-Maclaurin expansion and a new q-Leibniz formula were derived. In [148], [150], [203], [126] and [206]... |

153 |
On the structure of the topological phase of two-dimensional gravity
- Witten
- 1990
(Show Context)
Citation Context ...Verlinde (WDVV) hierarchy (10) is a remarkable system of partial differential equations determining deformations of 2−dimensional topological field theories, which was introduced in 1990 by Witten E. =-=[940]-=- and Dijkgraaf R., Verlinde E., Verlinde H. [228]. The WDVV equations consist of three groups of differential equations called the associativity equations,s36 THOMAS ERNST the normalisation equations,... |

138 |
Differential Forms with Applications to the Physical Sciences
- Flanders
- 1989
(Show Context)
Citation Context ... a one-soliton solution. The KdV hierarchy is connected with symmetric functions, Riemann surfaces, differential geometry, Hamiltonian structures, Poisson brackets and Schouten brackets [226], [100], =-=[299]-=-, [594]. We will now try to give a ’short’ introduction to the KdV hierarchy taken from [226, pp. 1-5] . The author ventures to offer one more monograph to the reader’s attention despite a rather grea... |

137 |
Theory of Groups of Finite Order
- Burnside
- 1911
(Show Context)
Citation Context ... finite groups [610]. The Mathieu groups were called ’sporadic’ for the first time in the book of Burnside, who noted that they ’would probably repay a closer examination than they have yet received’ =-=[137]-=-. The pioneer of the field in our century was Brauer, who made several crucial contributions to the classification problem [124]. Most of the finite simple groups, now called group of Lie type or Chev... |

136 |
Affine Hecke algebras and orthogonal polynomials. Cambridge Tracts
- Macdonald
- 2003
(Show Context)
Citation Context ...esult of [38] in the framework of Frenkel and Reshetikin [613]. In the same papers [36], [37] Aomoto also found a q-analogue of de Rham cohomology associated with Jackson integrals. In 1995 Macdonald =-=[595]-=- introduced nonsymmetric Macdonald polynomials, which are generalizations of Weyl’s formula for the characters of a compact Lie group. The Macdonald polynomials were related to certain induced represe... |

128 |
de Fériet, Fonctions hypergéométriques et hypersphériques. Polynômes d’Hermite. Gauthier-Villars
- Appell, Kampé
- 1926
(Show Context)
Citation Context ...nger’s equation, (11) Asymptotic theory of Hill’s equation, generalized Lamé equation, and generalized Weber equation, and (12) Theory of Lie algebras and Lie groups. In 1926 Appell & Kampé de Fériet =-=[42]-=- introduced some 2-variable hypergeometric series. One example is the Appell function ∞� (93) F2(a, b, b ′ ; c, c ′ ; x, y) = m,n=0 (a)m+n(b)m(b ′ )n m!n!(c)m(c ′ x )n m y n . In 1931 J. Horn (1867-19... |

127 |
The theory of group characters and matrix representations of groups
- Littlewood
- 1950
(Show Context)
Citation Context ...y the representations of the symmetric group become involved into solutions of difference equations. The relevant material on representations of the symmetric group can be found for example in [573], =-=[579]-=-, [661], [663], [802]. By the Frobenius character formula we obtain (537) ⎛ det(VY (λ)(r)) = det(V (r)) ⎝ � χλ(µ)c −1 where S ∗ |λ| µ∈S ∗ |λ| µ Sµ denotes the set of all conjugacy classes of S|λ|. ⎞ ⎠... |

124 |
de Vries G., On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves
- Korteweg
(Show Context)
Citation Context ...4]. Usually these equations admit a whole hierarchy of Poisson brackets [47, p. 135].sTHE HISTORY OF q-CALCULUS AND A NEW METHOD. 33 The KdV hierarchy (3) was derived in 1895 by Korteweg and de Vries =-=[536]-=- from the Navier-Stokes equation of fluid dynamics as a special limit to give a model of nonlinear wave motions of shallow water observed in a canal. They showed that the KdV equation admits a solitar... |

118 | Difference Equations and Inequalities - Agarwal - 2000 |

118 |
Partition function of the eight-vertex lattice model
- Baxter
- 1972
(Show Context)
Citation Context ...he method proposed by him, the famous Bethe Ansatz, has been applied successfully to other many-particle models in one-dimensional mathematical physics. In 1972 Rodney Baxter in his remarkable papers =-=[86]-=-- [87] (the results were announced by him in 1971 in [84]- [85]) gave a solution for the xyz model. He discovered a link between the quantum xyz model and a problem of two-dimensional classical physic... |

117 |
Methods of algebraic geometry in the theory of nonlinear equations
- Krichever
- 1977
(Show Context)
Citation Context ...]. It seems that all these equations are equivalent to linear systems of differential equations (for KdV Lax-pair). These linear systems contain a spectral parameter, which lives on a Riemann surface =-=[548]-=-, [657]. An excellent introduction to this is given in [839]. One of the most promising solution techniques for nonlinear differential equations rests on methods of algebraic geometry and leads to the... |

112 | Root systems and hypergeometric functions - Opdam - 1988 |

106 | Topics in noncommutative geometry - Manin - 1991 |

103 |
q-series: their development and application in analysis, number theory, combinatories, physics and computer algebra
- ANDREWS
- 1986
(Show Context)
Citation Context ...nd told him about a number of certain identities which he could not prove. Some of them were just corollaries of the Rogers-Ramanujan identities. The following citation is from Andrews excellent book =-=[31]-=-: (159) In 1913 Ramanujan astounded mathematicians by presenting them with a long list of remarkable propositions that he claimed to have discovered. On examination some of them proved old; many could... |

101 |
Exact results for a quantum many-body problem in one dimension
- Sutherland
(Show Context)
Citation Context ...chy [896]. (26) Kupershmidt’s equation [580]. (27) Sawada-Kotera equation [105], [944]. (28) BKP hierarchy [795],[582], [188]. (29) Calogero-Moser (CM) system. (30) Quantum Calogero-Sutherland system =-=[849]-=-, [850], [903]. (31) Maxwell-Bloch equation [359], [725]. (32) Ablowitz-Ladik hierarchy (ALH) [908]. (33) Davey-Stewartson (DS) equation [318], [555], [954]. (34) Kundu-Eckhaus equation [555]. (35) La... |

100 |
Theory of non-commutative polynomials
- Ore
- 1933
(Show Context)
Citation Context ...ors and algebraic-geometric constructions. Burchnall and Chaundy restricted themselves to operators with scalar-valued coefficients. The theory of non-commutative polynomials was explored by Ore 1933 =-=[701]-=- and by Jacobson 1937 [469]. In 1955 Weisner [926] explored the group-theoretic origin of certain generating functions. Let D = d dx and let f(D) = �n k=0 akDk be a general differential operator of de... |

99 |
A new class of integrable systems and its relation to solitons, Ann
- Ruijsenaars, Schneider
- 1986
(Show Context)
Citation Context ...ons [928]. (39) Hirota’s bilinear difference equation [775]. (40) Bullough-Dodd equations [930]. (41) Caudrey-Dodd-Gibbon equations [930], [969].s32 THOMAS ERNST (42) Ruijsenaars-Schneider (RS) model =-=[762]-=-, [763], [764], [765], [766]. (43) Veselov-Novikov hierarchy [873]. (44) Wess-Zumino-Novikov-Witten (WZNW) model [654], [334]. (45) Ashkin-Teller model [15]. (46) Fateev-Zamolodchikov model [15]. (47)... |

96 |
The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative
- WEISS
- 1983
(Show Context)
Citation Context ...an ordinary differential equation to have the Painlevé property is that there be a Laurent expansion which represents the general solution in a deleted neighbourhood of a pole. Recently, Weiss et.al. =-=[927]-=- have introduced the Painlevé property for partial differential equations. They applied the method to the soliton equations (KdV,KP) and found, in a remarkably straightforward manner, the well-known B... |

95 | The q-Schur algebra
- Dipper, James
- 1989
(Show Context)
Citation Context ...hur reduced the study of those polynomial representations of GL(n, F ), which are homogeneous of a given degree n to the representation theory of the Schur algebra SF (n, m). In 1989 Dipper and James =-=[229]-=- presented a q-Schur algebra. For a recent account on this see the 1998 book by Stephen Donkin [239], and [350], [46]. A q-analogue of homological algebra was presented in [244].s122 THOMAS ERNST 3.7.... |

91 | Harmonic analysis for certain representations of graded Hecke algebras - Opdam - 1995 |

90 |
The number of partitions of a set
- Rota
- 1964
(Show Context)
Citation Context ...ding polynomials as formal Dirichlet series. In 1965 Andrews [24] proved some identities for Mock theta functions by qcalculus technique. In 1978 Milne [631] used the finite operator calculus of Rota =-=[756]-=- to obtain a q-analogue of the Charlier polynomials and of Dobinski’s equality for the Bell numbers. The q-Bell numbers were expressed as the n:th moments of a q-Poisson distribution. A q-Stirling num... |

87 |
Lezioni di Geometria Differenziale v.II, parte
- Bianchi
- 1927
(Show Context)
Citation Context ...quation (1) [103, p. 137], which encodes the whole geometry of the pseudospherical surfaces. Moreover the method of construction of a new pseudospherical surface from a given one, proposed by Bianchi =-=[102]-=-, gives rise to the Bäcklund transformation for the SG equation [65]. The connection between geometry and integrable PDE’s became even deeper when Hasimoto [420] found the transformation between the e... |

87 |
Affine Lie Algebras and Quantum Groups
- Fuchs
- 1992
(Show Context)
Citation Context ... Tλ. The element (13) eY = mY m! hY is an idempotent in IY . According to the hook rule, � (lp − lr) (14) mY = m! where p<r p n� λj! j=1 (15) lk = λk + n − k. Remark 2. The q-hook rule was defined in =-=[332]-=-. ,s8 THOMAS ERNST We know that every element of Sm is uniquely the product of disjoint cycles. The lengths of these cycles form a partition of m and, in this way, the conjugacy classes of Sm are inde... |

87 |
Method for Solving the Korteweg-de Vries Equation
- Gardner, Greene, et al.
- 1967
(Show Context)
Citation Context ...or a long time books had not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this followed one single work by Gardner, Green, Kruskal and Miura 1967 =-=[337]-=- about the Korteweg-de Vries equation (KdV): ∂u (101) = 6u∂u ∂t ∂x + ∂3u ∂3x which, before that work, seemed to be merely an unassuming equation of mathematical physics describing waves in shallow wat... |

86 |
Solvability of groups of odd order
- Feit, Thompson
- 1963
(Show Context)
Citation Context ... exist. The modern classification race started with the work of Feit and Thompson in 1962, who proved that every nonabelian finite simple group has even order, or equivalently, contains an involution =-=[279]-=-. This work made feasible the tremendous classification project led primarily by Gorenstein, resulting, after two decades of work by a large group of mathematicians, in the classification theorem [360... |

85 |
Group theory and physics
- Sternberg
- 1994
(Show Context)
Citation Context ...to David Hilbert. Though the new quantum mechanics had been initiated only in 1925, already in 1926-27 the mathematician Hilbert in Göttingen gave lectures on quantum mechanics [935]. The Gruppenpest =-=[842]-=- (the pest of group theory) would last for three decades [935].sTHE HISTORY OF q-CALCULUS AND A NEW METHOD. 5 1.1. Partitions, generalized Vandermonde determinants and representation theory. The theor... |

81 | Collisions of Calogero–Moser Particles and an Adelic Grassmannian”, preprint - Wilson - 1996 |

80 |
q-Hypergeometric Functions and Applications
- Exton
- 1983
(Show Context)
Citation Context ...fferentiable at x. Example 1. (109) Dq(x α ) = xα − (qx) α (1 − q)x = xα (1 − q α ) x(1 − q) = {α}qx α−1 , α ∈ C. The formulae for the q-difference of a sum, a product and a quotient of functions are =-=[261]-=-: (110) Dq (u(x) + v(x)) = Dqu(x) + Dqv(x). (111) Dq (u(x)v(x)) = Dqu(x)v(x) + u(qx)Dqv(x). (112) � � u(x) Dq = v(x) v(x)Dqu(x) − u(x)Dqv(x) , v(qx)v(x) �= 0. v(qx)v(x) Applying the Taylor formula to ... |

78 |
On q-analogues of the quantum harmonic oscillator and the quantum group SUq(2
- Macfarlane
- 1989
(Show Context)
Citation Context ... Lie algebras is obtained. This R-matrix depends on the spectral parameter through trigonometric functions. The q-analogue of the harmonic oscillator was derived by Kuryshkin [561] (1980), Macfarlane =-=[596]-=- (1989), Biedenharn [104] (1989) and others. Biedenharn showed its connection to the SU(2)q algebra. In 1990 Soni [812] found q-analogues of some prototype Berry phase calculations. In 1990 Gray & Nel... |

77 | Some Conjectures for Root Systems - Macdonald - 1982 |

74 |
The theory of the double gamma function
- Barnes
- 1901
(Show Context)
Citation Context ... of this section is based on Ueno K. & Nishizawa M. 1997 [889], [890]. Multiple gamma functions were introduced by Barnes as an infinite product regularized by the multiple Hurwitz zeta function [72],=-=[73]-=-,[74],[75]. Hardy [416],[417] studied this function from the viewpoint of the theory of elliptic functions. Kurokawa [560] showed that multiple gamma functions play an essential role to express gamma ... |

72 | Affine root systems and Dedekind’s η-function - MacDonald - 1972 |