User blog comment:B1mb0w/J Function Sandpit J 4/@comment-5529393-20151014104445

You have a lot of values but you never give a definition for the function itself.

There isn't really a "standard" definition for fundamental sequences, even below epsilon_0, although I guess we've agreed upon one on this wiki. (It seems natural enough to define (a + w^(b+1))[n] = (a + w^b n) ). If you asked me for a set of rules below Gamma_0 to call standard, I suppose I would go:

if b is limit, phi(a, b)[n] = phi(a, b[n])

phi(0, b+1) [n] = phi(0, b) * n

phi(a+1, 0) [0] = 0

phi(a+1, 0) [n+1] = phi(a, phi(a+1, 0)[n])

phi(a+1, b+1) [0] = phi(a+1, b) + 1

phi(a+1, b+1) [n+1] = phi(a, phi(a+1,b+1) [n])

if a is limit, phi(a, 0) [n] = phi (a[n], 0)

if a is limit, phi(a, b+1) [n] = phi(a[n], phi(a, b) + 1)

But there are other ways you could go. One thing that bothers me about the above is that it defines epsilon_0 [0] = 0, so epsilon_0 [n] = w^^(n-1) rather than w^^n. we should define phi(a+1, 0)[0] = 1? I dunno.

Using the above rules, then f_{zeta_0}(2) = f_{epsilon_epsilon_0}(2), but if we make the change to phi(a+1, 0)[0] = 1, then f_{zeta_0} (2) = f_{epsilon_epsilon_1} (2).

Sorry if this answer was less decisive than you probably wanted.