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9j-coefficients and higher

Boekbijdrage - Hoofdstuk

The su(2) 3j-coefficients (or 3j-symbols), 6j-coefficients, 9j-coefficients and higher (referred to as 3nj-coefficients) play a crucial role in various physical applications dealing with the quantization of angular momentum. In this chapter, the hypergeometric expressions for these coefficients and their relations to discrete orthogonal polynomials is emphasized. We give a short summary of the relevant class of representations of the Lie algebra su(2), and discuss their tensor product. In the tensor product decomposition, the important Clebsch-Gordan coefficients appear. 3j-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 6j-coefficients) are defined. The explicit expression of a Racah coefficient as a hypergeometric series of type 4F3 and the connection with Racah polynomials and their orthogonality is given. 9j-Coefficients are then defined in the context of the tensor product of four representations. They are related to a discrete orthogonal polynomial in two. Finally, we consider the tensor product of (n+1) representations and generalized recoupling coefficients or 3nj-coefficients, determined by two binary coupling schemes.
Boek: Encyclopedia of special functions : the Askey-Bateman project : volume 2 : multivariable special functions
Series: Encyclopedia of Special Functions: the Askey-Bateman Project.
Pagina's: 402 - 419
ISBN:9780511777165
Jaar van publicatie:2020
Toegankelijkheid:Open