Exact computation and asymptotic approximations of 6<i>j</i> symbols: Illustration of their semiclassical limits

Abstract

Abstract This article describes a direct method for the exact computation of 3 nj symbols from the defining series, and continues discussing properties and asymptotic formulas focusing on the most important case, the 6 j symbols or Racah coefficients. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behavior is studied to account for some of the most relevant features, both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas, such as those due to Ponzano and Regge and Schulten and Gordon. Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner's reduced rotation matrices (or Jacobi polynomials) and harmonic oscillators (or Hermite polynomials). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010