  We use the numerical renormalization group to calculate the auxiliary
spectral functions of the $U=\infty$ Anderson impurity model. The slave--boson
and pseudo--fermion spectral functions diverge at the threshold with exponents
$\alpha_{b}$ and $\alpha_{f}$ given in terms of the conduction electron phase
shifts by the X--ray photoemission and the X--ray absorption exponents
respectively. The exact exponents obtained here depend on the impurity
occupation number, in contrast to the NCA results. Vertex corrections in the
convolution formulae for physical Green's functions are singular at the
threshold and may not be neglected in the Fermi liquid regime.
