What does "$(-1/2)_k$" mean in this expression related to incomplete elliptic integrals?

Question

I've found an approximation definition for an incomplete elliptic integral of the second kind which uses the notation $$ \left( -\frac{1}{2} \right)_k. $$

I haven't seen this notation before. Can someone explain this to me?

The complete formula, from the Wolfram documentation for the EllipticE() function, is shown below:

Answer 1

This is almost certainly the Pochhammer symbol. For $x \in \mathbb{C}$ and $k \in \mathbb{N}$, this is defined by the formula $$ (x)_k = x(x+1)(x+2)\dotsb (x+k-1) = \prod_{j=0}^{k-1} (x+j) = \frac{\Gamma(x+n)}{\Gamma(x)}, $$ where $\Gamma$ is the Gamma function (which generalizes the factorial function). This is also sometimes called the "rising factorial."

Note that this is not the best notation. Even the above linked MathWorld article makes this point:

The Pochhammer symbol $(x)_n$ ... for $n\ge 0$ is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power or ascending factorial. ... In combinatorics, the notation $x^{(n)}$, $\langle x\rangle_n$, or $x^{n}$ is used for the rising factorial, while $(x)_n$ or $x^n$ denotes the falling factorial. Extreme caution is therefore needed in interpreting the notations $(x)_n$ and $x^{(n)}$.

[emphasis mine; references removed for readability]

However, in the context of elliptic integrals (which are among the functions studied in the "theory of special functions" alluded to in the above quoted material), the correct interpretation is likely the rising factorial. Note that in the documentation linked in the question, one of the representations of the elliptic integral $E(z \mid m)$ is in terms of a hypergeometric series, which can be expanded in terms of the rising factorial.