On the structure of the canonical tensor operators in the unitary groups. III. Further developments of the boson polynomials and their implications

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Abstract
Three structurally distinct and explicit expressions are developed for the boson polynomials. The relationship of these polynomials to the representations of the general linear group and the Gel'fand‐Graev generalized beta functions is noted. A by‐product of these results is a new, closed expression for the irreducible representations of the symmetric group. Some similarities, as well as dissimilarities, between the boson polynomial forms and the canonical tensor operator forms are presented and discussed, the origin of these properties being traced to the similarities and distinctions between Wigner coefficients and Racah coefficients. One of the boson polynomial expressions is used to prove an important new relation in the Racah‐Wigner calculus: the identity of the set of extended projective coefficients to a subset of Racah coefficients. This relationship becomes one‐to‐one for SU(2) and establishes a pattern calculus for the Racah coefficients of angular momentum theory.