Triggered by the interesting finding of Andrew I open this thread for the further investigation of elementary superfunctions, i.e. functions that are expressible with elementary functions and operations such that

for a given elementary function .

Our first example is:

with a superfunction .

Now the has the same property as the . Hence is another superfunction of .

Indeed exists and is differentiable at .

But it does not exist at the other fixed point , because is not defined.

Edit: both are regular super-functions at fixed point 1. .

So if we are at polynomials , we can also give an elementary superfunction for , i.e. .

Because .

Edit: these are the regular super-exponentials at 1. .

Generally for Chebyshev polynomials, these are the polynomials such that - for example above we used -, we know already two elementary superfunctions of , these are and .

for a given elementary function .

Our first example is:

with a superfunction .

Now the has the same property as the . Hence is another superfunction of .

Indeed exists and is differentiable at .

But it does not exist at the other fixed point , because is not defined.

Edit: both are regular super-functions at fixed point 1. .

So if we are at polynomials , we can also give an elementary superfunction for , i.e. .

Because .

Edit: these are the regular super-exponentials at 1. .

Generally for Chebyshev polynomials, these are the polynomials such that - for example above we used -, we know already two elementary superfunctions of , these are and .