  We have computed the first 30 even-N moments for the double fermionic (nf^2)
part of the quark-to-gluon splitting function P_{gq} at the fourth order of
perturbative QCD via the renormalization of off-shell operator matrix elements.
From these results we have determined the all-N form, and hence the exact
x-space expression, using systems of Diophantine equations for its
coefficients. The dominant and subdominant leading small-x nf^2 contributions
to P_{gq}^(3)(x) are of the form 1/x ln x and ln^4 x, respectively; the leading
large-x term is ln^4 (1-x). The coefficient of the first of these is new, the
other two agree with results obtained before and thus provide checks of our
results.
