Typeset Kampé de Fériet function

You can use an array. You get staggered subscript and superscript by setting them to an empty subformula.

\documentclass{article}

\usepackage{amsmath}

\newcommand{\linefill}{% a variation on \rightarrowfill
  {-}\mkern-7mu
  \cleaders\hbox{$\mkern-2mu-\mkern-2mu$}\hfill
  \mkern-7mu{-}%
}

\begin{document}

\begin{equation}
F{}_{2:0:0}^{0:2:1}
  \left[
   \setlength{\arraycolsep}{0pt}% local assignment
   \begin{array}{c@{{}:{}}c@{;{}}c}
   \linefill & \dfrac{a+b+1}{2},\dfrac{a-b+1}{2} & \quad 1 \\[1ex]
   \dfrac{a+b+3}{2},\dfrac{a-b+3}{2} & \qquad\linefill & \linefill
   \end{array}
   \;\middle|\;
   x\dfrac{y^2}{4},\dfrac{y^2}{4}
 \right]
\end{equation}

\end{document}

Mixed use of arrays, \dfracs and \phantoms:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\[
  F \mathstrut_{2:0;0}^{0:2;1}
    \left[
      \begin{array}{@{} c @{:} c @{;} c @{;}}
        \dfrac{\phantom{\mu + \nu + 3, \mu - \nu + 3}}{\phantom{2}} & 
          \dfrac{\mu + \nu + 1}{2}, \dfrac{\mu - \nu + 1}{2} &
          1 \\
        \dfrac{\mu + \nu + 3}{2}, \dfrac{\mu - \nu + 3}{2} &
          \dfrac{\phantom{\mu + \nu + 1, \mu - \nu + 1}}{\phantom{2}} &
          \dfrac{\phantom{0}}{\phantom{0}}
      \end{array}
      a \dfrac{z^2}{4}, \dfrac{z^2}{4}
    \right]
\]

\end{document}