Contiguous relations, continued fractions and orthogonality

Gupta, Dharma P.; Masson, David R.

doi:10.1090/s0002-9947-98-01879-0

Cited by 43 publications

(27 citation statements)

References 21 publications

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“…Note also that the lattices Λ and Q E The bilinear form λ · µ on the Picard lattice Λ is interpreted as the intersection form of divisors λ, µ. In general, 31) corresponds to the class of curves on P…”

Section: Picard Latticementioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

123

250

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

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Section: Picard Latticementioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

123

250

View full text Add to dashboard Cite

show abstract

“…It seems desirable to classify all limit cases, along with their continuous relatives. Many known systems (see [AI,AV,GM,IM1,IM2,K3,P,R1,R2,R3,R4] for some candidates) should fit into this larger picture.…”

Section: Introductionmentioning

confidence: 99%

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Rosengren

2007

Ramanujan J

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Abstract. Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson's biorthogonal 10 W 9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations. IntroductionThe classical 6j-symbols were introduced by Racah and Wigner in the early 1940's [Rac, Wi]. Though they appeared in the context of quantum mechanics, they are natural objects in the representation theory of SL(2) that can be introduced from purely mathematical considerations. Wilson [W1] realized that 6j-symbols are orthogonal polynomials, and that they generalize many classical systems such as Krawtchouk and Jacobi polynomials. This led Askey and Wilson to introduce the more general q-Racah polynomials [AW1].The q-Racah polynomials belong to the class of basic (or q-) hypergeometric series [GR1]. Since the 1980's, there has been a considerable increase of interest in this classical subject. One reason for this is relations to solvable models in statistical mechanics, and to the related algebraic structures known as quantum groups.Kirillov and Reshetikhin [KR] found that q-Racah polynomials appear as 6j-symbols of the SL(2) quantum group, or quantum 6j-symbols. We mention that in the introduction to the standard reference [CP], three major applications of quantum groups to other fields of mathematics are highlighted. For at least two of these, namely, invariants of links and three-manifolds [Tu], and the relation to affine Lie algebras and conformal field theory [EFK], quantum 6j-symbols play a decisive role.The q-Racah polynomials form, together with the closely related Askey-Wilson polynomials, the top level of the Askey Scheme of (q-)hypergeometric orthogonal polynomials [KS]. One reason for viewing this scheme as complete is Leonard's 1991 Mathematics Subject Classification. 33D45, 33D80, 82B23.

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“…Consider Proposition 1 in the case of Wilson's basic hypergeometric biorthogonal rational functions. These functions depend on 6 parameters a, b, d, e, N, + and have the following explicit expression [9] R n (z)=C (1) n 10 W 9 (a; be &! , +be !…”

Section: When Biorthogonality Becomes Orthogonality?mentioning

confidence: 99%

“…, +be ! , d, e, aq N+1 , sq n&1 , q &n ; q), (7.10) upon the parameters (for details see [9]). The integer parameter N is the same as in our biorthogonality relation (4.26).…”

Section: When Biorthogonality Becomes Orthogonality?mentioning

confidence: 99%

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Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

Zhedanov

1999

Journal of Approximation Theory

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We present some general results concerning so-called biorthogonal polynomials of R II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel Darboux formula is obtained. Academic Press

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Contiguous relations, continued fractions and orthogonality

Cited by 43 publications

References 21 publications

Geometric aspects of Painlevé equations

Geometric aspects of Painlevé equations

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

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