Generalized Orthogonality and Continued Fractions

Ismail, Mourad E. H.; Masson, David R.

doi:10.1006/jath.1995.1106

Cited by 99 publications

(121 citation statements)

References 22 publications

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“…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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Section: Introductionmentioning

confidence: 99%

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

Rosengren

2007

Ramanujan J

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“…That the orthogonal systems corresponding to (1.3) and (1.4) involve rational functions rather than polynomials has been demonstrated by Ismail and Masson [17].…”

Section: Introductionmentioning

confidence: 86%

Contiguous relations, continued fractions and orthogonality

Gupta

Masson

1998

Trans. Amer. Math. Soc.

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Abstract. We examine a special linear combination of balanced very-wellpoised 10 φ 9 basic hypergeometric series that is known to satisfy a transformation. We call this Φ and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for Φ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's q-analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new q-series identities are also obtained. One is an important three-term transformation for Φ's which generalizes all the known two-and three-term 8 φ 7 transformations. Others are new and unexpected quadratic identities for these very-well-poised 8 φ 7 's.

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“…Taking again the initial condition R * 0 = 1 we arrive at the expression 10) where P n (z) is the same polynomial as in (2.6) and…”

Section: Gevp For Tri-diagonal Operators and Biorthogonal Rational Fumentioning

confidence: 99%

“…which belongs to a class of so-called R II recurrence relations considered in [10]. There is a relation of the polynomial P n (z) with the characteristic determinant ∆ n (z) of truncated matrices [20]:…”

Section: Gevp For Tri-diagonal Operators and Biorthogonal Rational Fumentioning

confidence: 99%

Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

Zhedanov¹

2005

JNMP

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We consider the generalized eigenvalue problem Aψ = λBψ for two operators A, B. Self-similar closure of this problem under a simplest Darboux transformation gives rise to two possible types of regular algebras of dimension 2 with generators A, B. Realization of the operators A, B by tri-diagonal operators leads to a theory of biorthogonal rational functions. We find the general solution of this problem in terms of the ordinary and basic hypergeometric functions. In special cases we obtain general Padé interpolation tables for the exponential and power function on uniform and exponential grids.

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Generalized Orthogonality and Continued Fractions

Cited by 99 publications

References 22 publications

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

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

Contiguous relations, continued fractions and orthogonality

Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

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