  We present the even-N moments N =< 20 of the fourth-order (N^3LO)
contribution P_{gq}^(3)(x) to the quark-to-gluon splitting function in
perturbative QCD. These moments, obtained by analytically computing off-shell
operator matrix elements for a general gauge group, agree with all known
results, in particular with the moments N =< 10 derived before from structure
functions in deep-inelastic scattering. Using the new moments and the available
endpoint constraints, we construct approximations for P_{gq}^(3)(x) which
improve upon those obtained from the lowest five even moments. The remaining
uncertainties of this function are now practically irrelevant at momentum
fractions x > 0.1. The resulting errors of the convolution of P_{gq} at N^3LO
with a typical quark distribution are small at x >~ 10^{-3} and exceed 1% only
at x ~< 10^{-4} for a strong coupling alpha_s = 0.2. The present results for
P_{gq}^(3)(x) should thus be sufficient for most collider-physics applications.
