𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

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

doi:10.1090/s0002-9947-1994-1264148-6

Cited by 67 publications

(65 citation statements)

References 39 publications

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“…[4], [12], the Al-Salam-Carlitz polynomials [2], the symmetrized version of polynomials of Berg-Valent ( [3]) leading to a Freud-like weight [5], and the q −1 -Hermite polynomials of Ismail and Masson [9]. If we introduce the coefficients of the orthonormal polynomials as…”

Section: In the Indeterminate Case The Smallest Eigenvalue λ N Of Tmentioning

confidence: 99%

Small eigenvalues of large Hankel matrices: The indeterminate case

Berg¹,

Chen²,

Ismail³

2002

MATH. SCAND.

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Section: In the Indeterminate Case The Smallest Eigenvalue λ N Of Tmentioning

confidence: 99%

Small eigenvalues of large Hankel matrices: The indeterminate case

Berg¹,

Chen²,

Ismail³

2002

MATH. SCAND.

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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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“…In the same way, from (2.8), { K(·,λ n ) 2 (Z) S n (λ)} n∈Z belongs to 2 (Z). As a consequence, by using the CauchySchwarz inequality in 10) we obtain that the series in (2.3) converges absolutely for each λ ∈ C. Finally, by proceeding as in (2.10), we have…”

mentioning

confidence: 87%

“…The corresponding sampling expansion can be written as a Lagrange-type interpolation series 10) where the convergence of the series is absolute and uniform on compact subsets of the complex plane. Finally, in [10] one can find an example, the q-Hermite polynomials, where all the necessary ingredients for the analytic discrete Kramer sampling theorem are explicity computed.…”

Section: Orthogonal Polynomials As Discrete Analytic Kramer Kernelsmentioning

confidence: 99%

On the analytic form of the discrete Kramer sampling theorem

Garcı́a

Hernández-Medina

Muñoz-Bouzo³

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The classical Kramer sampling theorem is, in the subject of self-adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm-Liouville problems. In this paper, a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.2000 Mathematics Subject Classification. Primary 94A20, 44A60.

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𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

Cited by 67 publications

References 39 publications

Small eigenvalues of large Hankel matrices: The indeterminate case

Small eigenvalues of large Hankel matrices: The indeterminate case

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

On the analytic form of the discrete Kramer sampling theorem

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