Consider the following function:
$f(a,b,c)=\frac{1}{[1 + 2(a + c)][1 + 2(b + c)]}\,_3F_2 \left[ \begin{array}{l} 1, 1/2-c, -c \\ 1/2 - a - c, 1/2 - b - c \end{array} ; 1\right] $
with $a,b,c \in \mathbb{N}$.
I can be checked (for instance on Wolfram Mathematica) that $f(a,b,c)$ is a symmetric in its arguments. This is evident for $a \leftrightarrow b$ but it is far from obvious, for instance, when it comes to $a \leftrightarrow c$.
Suppose you don't know $f$ is symmetric. Is it possible to rewrite $f$ in a form where the symmetry property is evident for all possible exchanges?
Thanks