User blog comment:Moooosey/dco + decord sequence/@comment-43798125-20191014193224/@comment-35470197-20191015115817

Hmm...? I might not be understanding what you are asking, because it is the same as the rule for w^a[n]. (Remember that phi_0(x) coincides with w^x.)

For example, does the following definition of phi_0(a)[n] for a > e_0 make sense for you? Completely the same as the rule for w^a. Well, I personally recommend not to use w^a and e_0, because you have phi_0(a) and phi_1(0). Employing unnecessary functions and constants make the rules more complicated.
 * 1) If a is a successor ordinal b+1, then phi_0(a)[n] = phi_0(b)×n.
 * 2) If a is a limit ordinal such that a < phi_0(a), then phi_0(a)[n] = phi_0(a[n]).