Semiclassical limits of extended Racah coefficients

We explore the geometry and asymptotics of extended Racah coefficients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike. The asymptotic formulas derived for the extension are found to have a similar form to the standard Ponzano–Regge asymptotic formulas for the SU(2) 6j symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity.