Now I indeed had a look at

Lets starting with his theorem:

What however really bothers me, that it seems not to be true:

Let . This is a feasible function for theorem 2, with and .

Now I looked at the (unqiue) half iterate

which should be entire too, for comparison some members of its series:

and tested convergence at (an entire function has an infinite radius of convergence, so it should converge for every ) and what did I find? Divergence!

So it seems this proof is also not reliable...

Boy that shakes my trust in professional mathematics.

Quote:[1] P. L. Walker, A class of functional equations which have entire solutions, Bull. Austral. Math. Soc. 38 (198, no. 3, 351-356but things become more complicated!

Lets starting with his theorem:

Quote:Theorem 2. Let be an entire function of the form , where for all , and either (i) or (ii) .Where (2) is . The also mentioned theorem 1 and sequence does not matter yet.

Then the sequence defined in Theorem 1 converges uniformly on every to a function which is an entire non-constant solution of (2).

What however really bothers me, that it seems not to be true:

Let . This is a feasible function for theorem 2, with and .

Now I looked at the (unqiue) half iterate

which should be entire too, for comparison some members of its series:

and tested convergence at (an entire function has an infinite radius of convergence, so it should converge for every ) and what did I find? Divergence!

Quote:0, 5.0, 17.50000000, -13.75000000, 142.5000000, -834.0625000, 5757.734375, -38187.57812, 222737.7149, -830118.7301, -3591005.876, 123831803.7, -1672085945.

So it seems this proof is also not reliable...

Boy that shakes my trust in professional mathematics.