9 j-Coefficients and Higher

Abstract

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

Symmetry of terminating series representations of the Askey-Wilson polynomials

“…Further continuation of our study of the mapping properties of these functions was made clear by previous and our work on the group theoretic description of the transformation properties of these functions (see e.g., [12,16] and Propositions 15,17 below). This work in this present paper provides a framework for future work on the symmetry analysis of terminating basic hypergeometric functions which is more complicated than that for the nonterminating case [15] and that it is not surprising that the classes of terminating basic hypergeometric functions are not connected by the known nonterminating transformations (see Figures 1,2,3 below). In this paper, for the first time, we present the full symmetry structure of the terminating 8 W 7 representations for the Askey-Wilson polynomials and a detailed connection with the terminating balanced 4 φ 3 representations.…”

“…Both expressions (29), (30), map to the basic hypergeometric representation (17), except with (30), one has θ → −θ. For the 4 φ 3 expressions under the standard map (43), the expression (31) maps to (16) and the expression (32) maps to (15). Similarly for the 8 W 7 expressions using (43), then (22), (23) (θ → −θ) map to (21); (24) maps to (20); ( 27), (26) (θ → −θ) maps to (18); and (25), (28) (θ → −θ) maps to (19).…”