U( n) Wigner coefficients, the path sum formula, and invariant G-functions

We prove the path sum formula for computing the U( n) invariant denominator functions associated to stretched U( n) Wigner operators. A family of U( n) invariant polynomials G [ λ] ( n) is then defined which generalize the μ G q ( n) polynomials previously studied. The G [ λ] ( n) polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the μ G q ( n) polynomials. We also give a direct proof of the important transposition symmetry for the G [ λ] ( n) polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U( n) Wigner operators.