The triple sum formulas for 9j coefficients of SU(2) and uq(2)

Alisauskas, S

doi:10.1063/1.1312198

Cited by 9 publications

(37 citation statements)

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“…The proof of (11a) follows by comparing equations (4.3c) and (4.3e) of [18], making appropriate relabellings, and using the same rational expression argument as in the proof of Theorem 1. In a similar way, (11b) follows from (4.3b) and (4.4c) of [18].…”

Section: Kampé De Fériet Series With Two Negative Integers As Parmentioning

confidence: 99%

“…The first formula, (12a), follows by comparing expressions (4.3a) and (4.3d) of [18], and using the rational expression argument. The second formula is derived using (11a) and…”

Section: Kampé De Fériet Series With Two Negative Integers As Parmentioning

confidence: 99%

“…In [15], Rosengren deduced the triple sum series for 9-j coefficients (of su(1, 1) rather than of su (2)) based upon the use of coupling kernels; in [16], he showed that the same formula can be deduced starting from the classical expansion of the 9-j coefficient in terms of 6-j coefficients and performing Dougall's summation formula [17] for a very well-poised 4 F 3 (−1) series. In a recent paper [18], Ališauskas realized that this technique can be applied for several distinct expansions of the 9-j coefficient in terms of 6-j coefficients. Thus he obtained seven different triple sum formulas for the 9-j coefficient of su (2).…”

Section: Introductionmentioning

confidence: 99%

“…Since they stand for angular momenta, all the arguments in (4) are nonnegative integers or half-integers. In fact, in [18], the q-analogues of such 9-j coefficients are considered, but here we first treat the classical case (q = 1).…”

Section: Introductionmentioning

confidence: 99%

“…In the paper of Ališauskas [18], the emphasis is on the q-9-j coefficients, i.e. the 9-j coefficients of su q (2).…”

Section: Introductionmentioning

confidence: 99%

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Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

Lievens

Jeugt

2001

Journal of Mathematical Physics

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In a recent paper, Ališauskas deduced different triple sum expressions for the 9-j coefficient of su(2) and su q (2). For a singly stretched 9-j coefficient, these reduce to different double sum series. Using these distinct series, we deduce a set of new transformation formulas for double hypergeometric series of Kampé de Fériet type and their basic analogues. These transformation formulas are valid for rather general parameters of the series, although a common feature is that all the series appearing here are terminating. It is also shown that the transformation formulas deduced here generate a group of transformation formulas, thus yielding an invariance group or symmetry group of particular double series.Running title : Transformations for double series.

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Section: Kampé De Fériet Series With Two Negative Integers As Parmentioning

confidence: 99%

“…The first formula, (12a), follows by comparing expressions (4.3a) and (4.3d) of [18], and using the rational expression argument. The second formula is derived using (11a) and…”

Section: Kampé De Fériet Series With Two Negative Integers As Parmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In the paper of Ališauskas [18], the emphasis is on the q-9-j coefficients, i.e. the 9-j coefficients of su q (2).…”

Section: Introductionmentioning

confidence: 99%

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Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

Lievens

Jeugt

2001

Journal of Mathematical Physics

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9 j-Coefficients and Higher

Jeugt¹

2020

Encyclopedia of Special Functions: The Askey-Bateman Project

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9 j-Coefficients and higher Joris Van der Jeugt 12.1 Introduction 3 j-Coefficients (or 3 j-symbols), 6 j-coefficients, 9 j-coefficients and higher (referred to as 3n jcoefficients) play a crucial role in various physical applications dealing with the quantization of angular momentum. This is because the quantum operators of angular momentum satisfy the su(2) commutation relations. So the 3n j-coefficients in this chapter are 3n j-coefficients of the Lie algebra su(2). For these coefficients, we shall emphasize their hypergeometric expressions and their relations to discrete orthogonal polynomials. Note that 3n j-coefficients can also be considered for other Lie algebras. For positive discrete series representations of su(1, 1), the 3n j-coefficients carry different labels but have the same structure as those of su(2) [29,37], and the related orthogonal polynomials are the same. For other Lie algebras, the definition of 3 j-coefficients (i.e., coupling coefficients or Clebsch-Gordan coefficients related to the decomposition of tensor products of irreducible representations) is more involved, since in general multiplicities appear in the decomposition of tensor products [35,11]. Note that there is also a vast literature on the q-analogues of 3n j-coefficients in the context of quantum groups or quantized enveloping algebras: for the quantum universal enveloping algebra U q (su(2)), the 3 j-and 6 j-coefficients are straightforward q-analogues of those of su(2), and the related discrete orthogonal polynomials are the corresponding q-orthogonal polynomials in terms of basic hypergeometric series [21,23,1,2].Here, we shall be dealing only with 3n j-coefficients for su(2). First, we give a short summary of the relevant class of representations of the Lie algebra su(2). An important notion is the tensor product of such representations. In the tensor product decomposition, the important Clebsch-Gordan coefficients appear. 3 j-Coefficients are proportional to these Clebsch-Gordan coefficients. We give some useful expressions (as hypergeometric series) and their relation to Hahn polynomials. Next, the tensor product of three representations is considered, and the relevant Racah coefficients (or 6 j-coefficients) are defined. The explicit expression of a Racah coefficient as a hypergeometric series of 4 F 3 -type and the connection with Racah polynomials and their orthogonality is given. 9 j-Coefficients are defined in the context of the tensor product of four representations. They are related to a discrete orthogonal polynomial in two variables (but no expression as a hypergeometric double sum is known). Finally, we

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Covariance of the galaxy angular power spectrum with the halo model

Lacasa

2018

A&A

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As the determination of density fluctuations becomes more precise with larger surveys, it becomes more important to account for the increased covariance due to the non-linearity of the field. Here I have focussed on the galaxy density, with analytical prediction of the non-Gaussianity using the halo model coupled with standard perturbation theory in real space. I carried out an exact and exhaustive derivation of all tree-level terms of the non-Gaussian covariance of the galaxy C , with the computation developed up to the third order in perturbation theory and local halo bias, including the non-local tidal tensor effect. A diagrammatic method was used to derive the involved galaxy 3D trispectra, including shot-noise contributions. The projection to the angular covariance was derived in all trispectra cases with and without Limber's approximation, with the formulae being of potential interest for other observables than galaxies. The effect of subtracting shot-noise from the measured spectrum is also discussed, and does simplify the covariance, though some non-Gaussian shot-noise terms still remain. I make the link between this complete derivation and partial terms which have been used previously in the literature, including supersample covariance (SSC). I uncover a wealth of additional terms which were not previously considered, including a whole new class which I dub braiding terms as it contains multipole-mixing kernels. The importance of all these new terms is discussed with analytical arguments. I find that they become comparable to, if not bigger than, SSC if the survey is large or deep enough to probe scales comparable with the matter-radiation equality k eq . A short self-contained summary of the equations is provided in Section 9 for the busy reader, ready to be implemented numerically for analysis of current and future galaxy surveys.

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The triple sum formulas for 9j coefficients of SU(2) and uq(2)

Cited by 9 publications

References 37 publications

Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

9 j-Coefficients and Higher

Covariance of the galaxy angular power spectrum with the halo model

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