1968Journal of Mathematical PhysicsRequires access

Quasibinomial Representations of Clebsch-Gordan Coefficients. II. ``Negative'' Representations

S. M. Razaullah Ansari

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Abstract

New quasibinomial forms are derived from the quasibinomial forms given previously by making use of both positive and negative generalized powers. They turn out to be a new representation of the Wigner-type unsymmetrical formulas of Clebsch-Gordan coefficients for angular momenta. Consequently, formulas of Racah, Majumdar, and Shimpuku are deduced as special cases. Rules to construct a square symbol are given from which all these ``negative'' quasibinomial representations or, more precisely, expansions can be read off directly. Thus, a unified treatment of both symmetrical and unsymmetrical formulas of Clebsch-Gordan coefficients is thereby accomplished.

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New quasibinomial forms are derived from the quasibinomial forms given previously by making use of both positive and negative generalized powers. They turn out to be a new representation of the Wigner-type unsymmetrical formulas of Clebsch-Gordan coefficients for angular momenta. Consequently, formulas of Racah, Majumdar, and Shimpuku are deduced as special cases. Rules to construct a square symbol are given from which all these ``negative'' quasibinomial representations or, more precisely, expansions can be read off directly. Thus, a unified treatment of both symmetrical and unsymmetrical formulas of Clebsch-Gordan coefficients is thereby accomplished.

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Available abstract

New quasibinomial forms are derived from the quasibinomial forms given previously by making use of both positive and negative generalized powers. They turn out to be a new representation of the Wigner-type unsymmetrical formulas of Clebsch-Gordan coefficients for angular momenta. Consequently, formulas of Racah, Majumdar, and Shimpuku are deduced as special cases. Rules to construct a square symbol are given from which all these ``negative'' quasibinomial representations or, more precisely, expansions can be read off directly. Thus, a unified treatment of both symmetrical and unsymmetrical formulas of Clebsch-Gordan coefficients is thereby accomplished.

Key concepts: Clebsch–Gordan coefficients, Symbol (formal), Mathematics, Irreducible representation, Pure mathematics, Representation (politics), Algebra over a field, Construct (python library)

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