  We extend the threshold resummation exponents G^N in Mellin-N space to the
fourth logarithmic (N^3LL) order collecting the terms alpha_s^2 (alpha_s ln
N)^n to all orders in the strong coupling constant as. Comparing the results to
our previous three-loop calculations for deep-inelastic scattering (DIS), we
derive the universal coefficients B_q and B_g governing the final-state jet
functions to order alpha_s^3, extending the previous quark and gluon results by
one and two orders. A curious relation is found at second order between these
quantities, the splitting functions and the large-angle soft emissions in
Drell-Yan type processes. We study the numerical effect of the N^3LL
corrections using both the fully exponentiated form and the expansion of the
coefficient function in towers of logarithms.
