Im looking for a real-entire function $f(z)$ such that for any complex $z$ :
$1) $$f(z+p) =f(z)$
With $p$ a nonzero real number.
$2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than $50$ % of (signs of the) $a_n$ are $0$.
Thus let $f_n(z)$ be the truncated Taylor expansion of $f(z)$ of degree $n$. Let $T(n)$ be the amount of zero (signs in the) coefficients of the polynomial $f_n(z)$.
Then $\lim_{n -> +\infty} \dfrac{T(n)}{n} > \frac{1}{2}$.
$3)$ $f(z)$ is nonconstant.
Is such a function $f(z)$ possible ?
Related :
Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?
$$EDIT$$
I know that if $m$ is a positive integer different from $0,1,2,4$ and $g(z)$ is a nonconstant real-entire periodic function then $g(z^{\frac{1}{m}})$ is not entire. This rules out some periodic patterns in the derivatives as a potential solution to the question.
For instance that implies the pattern $[0,0,+,0,0,-]$ cannot occur.
Notice the pattern $[0,+,0,-]$ can occur and does because $[cos-1](\sqrt x)$ is entire.
I wanted to share this because it might help or inspire someone.
