Identity to show that a particular expression with hypergeometric functions is odd

I am working on a physics problem that results in the expression

$$f(x) = \frac{ _2F_1\left( \begin{matrix} \begin{matrix} a & b \end{matrix} \\ \frac{a+b}{2} + x \end{matrix} \hspace{6pt} \frac{1}{2} \right) }{ _2F_1\left( \begin{matrix} \begin{matrix} a - 1 & b \end{matrix} \\ \frac{a+b}{2} + x \end{matrix} \hspace{6pt} \frac{1}{2} \right) } - 1$$

For physical reasons, I know that $f(x)$ has to be odd as a function of $x$ (but note that $x$ is not the usual argument of the hypergeometric function here). This also seems to be true numerically.

Main question: Is there a simple hypergeometric identity that shows that $f(x)$ is odd in $x$?

• Obviously, you can show this by actually doing a series expansion in $x$, but this is cumbersome.
• In my particular case, $a$ happens to be a negative integer, and $b$ happens to be a positive integer. I don't know if this is relevant (i.e., whether $f(x)$ would still be odd if they were not integers or if they both had the same sign). However, I suspect that this will probably be true for any $a$ and $b$, since I don't think I have used the fact that $a$ and $b$ are integers at any point in the derivation.
• By bringing the $1$ into the fraction and applying a contiguous relation, you can re-cast this into a lot of different forms. Perhaps the simplest is to say that $g(x)$ is odd in $x$, where $g(x)$ is given by
$$g(x) = \left.\frac{d}{dz}\right|_{z=\frac{1}{2}} \log {_2}F_1\left( \begin{matrix} \begin{matrix} a - 1 & b \end{matrix} \\ \frac{a+b}{2} + x \end{matrix} \hspace{6pt} z \right)$$
• It seems like the easiest way to prove something like this would be if there was an identity (perhaps a quadratic transformation) that provided a relationship between $${_2}F_1\left( \begin{matrix} \begin{matrix} a & b \end{matrix} \\ c \end{matrix} \hspace{6pt} \frac{1}{2} \right)$$ and $${_2}F_1\left( \begin{matrix} \begin{matrix} a & b \end{matrix} \\ a + b - c \end{matrix} \hspace{6pt} \frac{1}{2} \right)$$ I have not been able to find anything along these lines. If anyone has a copy of Erdelyi's book with all the identities and is willing to look for one, I'd be very appreciative.