twice a superfunction

[MphLee]

i have been thinking about the following often :

how and when is a function a 'superfunction for two functions' and what degrees of freedom do we have ?

for instance

assume the following equation

f(x+1) = A(f(x))
f(x+i) = B(f(x))

Then the A and B always commute. I think that this is the criterion.

But maybe the the fact that they commute is too general maybe.

example

\( A=h^{\circ s} \)
\( B=h^{\circ t} \)

and

\( A\circ B=B \circ A \)
the functions commute but their superfunction (\( H \)) should be defined with

\( H(x+s) = A(H(x)) \)
\( H(x+t) = B(H(x)) \)

then we have \( h(x)=H(1+H^{-1}(x)) \).

But if we don't know such \( s \), \( t \) and \( h \) and we only know that A and B commute how can we know that \( f \) exist?

\( f(x+1) = A(f(x)) \)
\( f(x+i) = B(f(x)) \)


By the way I don't think that we can define a new superfunction from two functions A and B when they don't commute... if it is possible is really weird and interesting...

Have you found some example where we dont need that they commute and the superfunction exist?

PS: I replied to your private message.