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))

and A and B are somewhat related : they commute of course and satisfy some equation ( e.g. A = sin(log(B)) )

i assume there is no freedom for f(z) apart from choosing f(0).

( once again , riemann mapping theorem and double periodic functions convinced me of that )

but its not so clear when we have a solution and when not.

also , could this be a usefull uniqueness criterion ?

its seems we need at least 3 criterions for a solution to exist :

1) A and B commute
2) A and B share the same fixpoints
3) its clear the superfunction of A and B is nonparadoxal at complex oo.

maybe 4 : 4) the superfunction becomes periodic or semi-periodic near complex oo. ( although that might follow from 3 )

regards

tommy1729

\( A(x)=f(f^{\circ -1}(x)+1) \)
and
\( B(x)=f(f^{\circ -1}(x)+i) \)

but this doesn't mean that \( A^{\circ i}(x)=B(x) \) should hold always?