Min (f(x) / x^n) = 1/n!

My intuition suggests f(x) ~ exp(x) sqrt(x+1) / sqrt(2 pi).

Or in other notation S9^[-1] (exp(x)) ~ O( exp(x) sqrt(x) ).

More general S9^[r] (exp(x)) ~ O( C^r exp(x) x^(-r/2) ).

Also wondering about lim Gauss^[+oo](any(x)) = ?? If the limit even exists !?

I could not help noticing the resemblance to the semi-derivative of exp and S9(exp(x)). Coincidence ? Or does the semi-derivative play a role in fake function theory ?

Does for Large n,r :

a_n® a_n(-r) ~ gaussian a_n ?

( when the gaussian is good )

Regards

Tommy1729

My intuition suggests f(x) ~ exp(x) sqrt(x+1) / sqrt(2 pi).

Or in other notation S9^[-1] (exp(x)) ~ O( exp(x) sqrt(x) ).

More general S9^[r] (exp(x)) ~ O( C^r exp(x) x^(-r/2) ).

Also wondering about lim Gauss^[+oo](any(x)) = ?? If the limit even exists !?

I could not help noticing the resemblance to the semi-derivative of exp and S9(exp(x)). Coincidence ? Or does the semi-derivative play a role in fake function theory ?

Does for Large n,r :

a_n® a_n(-r) ~ gaussian a_n ?

( when the gaussian is good )

Regards

Tommy1729