Searching for an asymptotic to exp[0.5]

[tommy1729]

Conjecture T0 :

Let \( f(z) = a_0 + a_1 z + a_2 z^2 + ... \) be a real entire function where \( a_i > 0 \).

If for all \( n \) we have \( a_{n+1} < a_n/n \) then \( f(z) \) is stable.

This has probably been proved already.

(edit)

More specificly :

Conjecture T1 :

Let \( f_1(z) = A_0 + A_1 z + A_2 z^2 + ... \) be a real entire function where \( 0<A_i<1 \).

Let \( f_2(z) = B_0 + B_1 z + B_2 z^2 + ... \) be a real entire function where \( B_i >= 0 \) such that at least \( B_i \) is not equal to \( 0 \) for 2 values of \( i \).

Let \( f_3(z) = C_0 + C_1 z + C_2 z^2 + ... \) be a real entire function where \( C_i = [f_2({A_i}^{-1})]^{-1} \).

Then if \( f_1(z) \) is stable , then so is \( f_3(z) \).




regards

tommy1729