This LaTeX:
G_{p,q}^{m,n} \left(\begin{matrix}a_1, \dots, a_n & a_{n+1}, \dots, a_p \\
b_1, \dots, b_m & b_{m+1}, \dots, b_q
\end{matrix} \middle| z \right).
does not render at https://khan.github.io/KaTeX/. It works fine in MathJax and regular LaTeX.
The error in the console is
katex.min.js:1 Uncaught Error: KaTeX parse error: Expected & or \\ or \end at position 53: in{matrix}a_1, ̲\dots, a_n & a(…)
However, I can't figure out how to reduce it to an expression that works. It seems it does not like the matrix (I don't know why). If I put the matrix by itself it still fails. Additionally, it doesn't support \middle (even \left(x \middle | y \right) fails with katex.min.js:1 Uncaught Error: KaTeX parse error: Expected '\right', got '\middle' at position 29: \left(x \middle̲ | y \right)}(…)).
Looks like there's two issues, which can both be worked around for now:
\dots: it depends on observing the environment that it's in, so it's non-trivial to implement correctly. This can be replaced with \ldots, since that's what \dots would turn into in this case.\middle: I didn't even know that \middle existed! You can approximate it by doing something like \bigg| or \Big| in your case.So, with those two things fixed,
G_{p,q}^{m,n} \left(\begin{matrix}a_1, \ldots, a_n & a_{n+1}, \ldots, a_p \\
b_1, \ldots, b_m & b_{m+1}, \ldots, b_q
\end{matrix} \bigg| z \right).
ends up rendering fine. :)

Hopefully this is helpful! I'll file some issues for the missing features.
I didn't notice it was using \dots instead of \ldots. Perhaps there should be a third issue about the error message. If I replace \dots with a fake command, like \dts, LaTeX gives me this message:
! Undefined control sequence.
l.131 ...{p,q}^{m,n} \left(\begin{matrix}a_1, \dts
, a_n & a_{n+1}, \dots, a_...
! ==> Fatal error occurred, no output PDF file produced!
And MathJax renders the whole equation with a red \dts. The KaTeX error message doesn't indicate what went wrong.
At any rate, \dots doesn't sound like something you should ever use.
Also, it's funny, 5 years ago I reported \middle not working to MathJax, and they gave a similar response.
Most helpful comment
Also, it's funny, 5 years ago I reported \middle not working to MathJax, and they gave a similar response.