The double fermionic contribution to the four-loop quark-to-gluon splitting function

We have computed the first 30 even-N moments for the double fermionic (n(f)(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 n(f)(2) contributions to P-gq((3)) (x) are of the form x(-1) 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.