In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions:
$$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$
where $b>0$,$n\in\mathbb{N}$, $$u_k(z,n,b)=g_k(z,n,b)+\overline{g_k(\overline{z},n,b)}\tag{2}$$ $$g_k(z,n,b)=c_k(n) \frac{\sin\left(b(z-i(k+1/4))\right)}{b(z-i(k+1/4))}\tag{3}$$
$$c_k(n)=\pi^k a_k H_{n,-2k}\tag{4}$$ $$a_k=\frac{2k+1}{(k-1)!},k\in\mathbb{N}\tag{5}$$ And $H_{n,m}$ is the Harmonic number of order $m$, $$H_{n,m}=\sum_{k=1}^n k^{-m}.\tag{6}$$
Our goal is to prove that
Proposition 1 $f_m(z,n,b)$ as a function of z, has only real zeros.
One way we are trying to achieve this goal is to apply a theorem by Polya:
Theorem 2 (Polya [1]):
Assume (1) $G(z)$ is an entire function of genus 0 or 1 and all the zeros of $G(z)$ are in the half plane of $Im(z)>0$, i.e., $G(z)$ belongs to Hermite-Biehler class of entire functions;
(2) $$h(z)=\frac{G(z)}{\quad\overline{G(\overline{z})}\quad},|h(z)|<1,\qquad Im(z)>0\tag{6}$$ Then the function $G(z)+\overline{G(\overline{z})}$ has only real zeros.
We can readily show that $b(z-ia)^{-1}\sin(b(z-ia))$ with $a,b>0$ has the same properties as $G(z)$ in Theorem 1 has. Thus $u_k(z,n,b)$,$k\in\mathbb{N}$ are real-rooted.
Thus for any $n,m\in\mathbb{N},b>0$, $f_m(z,n,b)$ is a linear combination of real rooted entire function of genus 0. In addition, numerical results shows that sequence $c_k$ is log-concave (this is something we think that can be proved).
Any thoughts, suggestions, references on how to prove Proposition 1 are welcome.
[1] George Polya, Bemerkunguber die Integraldarstellung der Riemannsche ξ-Funktion, Acta Math. 48 (1926), 305–317