Unified Closed-Form Representations and Generating Functionals for SU(2) 3n-j Recoupling Coefficients

We present a unified framework for computing SU(2) 3nj recoupling coefficients through five complementary mathematical representations: hypergeometric product formulas, uniform single-sum expressions, finite recurrence relations, universal generating functionals, and arbitrary-valence matrix elements. This work establishes the first truly closed-form expressions for arbitrary trivalent graphs, validated through rigorous cross-verification against symbolic computation and high-precision reference datasets. Applications span quantum physics from atomic structure to loop quantum gravity.