3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

Abstract

In the tensor product of n+1 positive discrete series representations of su(1,1), a coupled basis vector can be described by a certain binary coupling tree. To every such binary coupling tree, polynomials Rl(k)(x) and Rl(k)(x) are associated. These polynomials are n-variable Jacobi and continuous Hahn polynomials, and are orthogonal with respect to a weight function. The connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj-coefficients of su(1,1).