Upon answering a question about an equivalence of two binomial sums I have noted that a naturally appearing function has some interesting properties.
Consider the function: $$ f(m,n_1,n_2;z)=\frac{1}{c+z}\sum_{k\ge0}\frac{\binom{n_1}{k}\binom{n_2}{k}}{\binom{m}{k}\binom{c+z-1}{k}}\tag{1}, $$ where $m,n_1,n_2$ are integer numbers $m\ge n_1,n_2\ge0$ and $z$ is a complex number (which can be treated in the context of question also as real). Besides the explicit symmetry between $n_1$ and $n_2$ the function has a "hidden" one, which can be revealed by appropriate choice of the argument shift $c$.
Introducing the following shortcuts: $$ n_\text{min}=\max(n_1,n_2),\quad n_\text{max}=\min(n_1+n_2,m), $$ the constant $c$ and the parameter $N$, which can be identified as the number of zeros of the function $f$, are defined as: $$ c=n_1+n_2-\frac{n_\text{min}+n_\text{max}}{2},\quad N=n_\text{max}-n_\text{min}. $$
With this choice of $c$ the following symmetry property holds: $$f(m,m-n_1,n_2;z)=-f(m,n_1,n_2;-z).$$
The function $f$ is obviously rational and can be represented as $$ f(m,n_1,n_2;z)=\frac{P_{m,n_1,n_2}(z)}{Q_{m,n_1,n_2}(z)}, $$ where $P(z)$ and $Q(z)$ are polynomials with integer coefficients having no common roots. In general: $$ P(z)=\sum_{k=0}^{N}A_k z^k,\quad Q(z)=A_{N}\prod_{k=0}^{N}(z+\frac{N}{2}-k), $$ where dependence of $A_k$ on $m,n_1,n_2$ is assumed. Observe that the poles of the function are symmetric with respect to $z=0$.
The following statements are by numerical evidence valid:
- $P(z)$ is irreducible over $\mathbb{Z}$ unless $n_1$ is odd, $m=2n_2$, in which case $P(z)$ is trivially divisible by $z$.
- All roots of $P(z)$ are real and distinct. The poles and zeros of $f(z)$ alternate: $z_0<\zeta_1<z_1<\dots<z_{N-1}<\zeta_{N}<z_{N}$.
In view of symmetry the relation 1. is valid by interchange of indices $n_1\leftrightarrow n_2$ as well. In 2. $z_k=-\frac{N}{2}+k$ and $\zeta_k$ are roots of $Q(z)$ and $P(z)$, respectively.
Interesting is also the fact that $f(2n,n,1;z)=\frac{\psi(z+3/2)-\psi(z-1/2)}{2}$, where $\psi(z)$ is polygamma function.
Can the properties be proved? Can a closed form expression for the roots of $P(z)$ be given?