Indefinite Integration of the Gamma Integral
and Related Statistical Applications
4.6 Hypergeometric Function
The hypergeometric function is the core of the cumulative distribution function of Student's t distribution
\begin{equation*}
F\left( x;\nu \right)=\frac{1}{2}+x\Gamma \left( \frac{\nu +1}{2} \right)\cdot \frac{{}_{2}{{F}_{1}}\left(\tfrac{1}{2},
\tfrac{\nu +1}{2};\tfrac{3}{2},\tfrac{-{{x}^{2}}}{\nu } \right)}{\sqrt{\pi \nu }\Gamma \left( \frac{\nu }{2} \right)},
\end{equation*}
where $\nu >0$ (degree of freedom) and $x\in \left( -\infty ,\infty \right)$
(Johnson & Kotz, 1970, p.96).
With a few steps, we can work out $_{2}{{F}_{1}}\left( \tfrac{1}{2},
\tfrac{\nu +1}{2};\tfrac{3}{2},\tfrac{-{{x}^{2}}}{\nu } \right)$ in the following form:
\begin{multline*}
_{2}{{F}_{1}}\left( \tfrac{1}{2},\tfrac{\nu+1}{2};\tfrac{3}{2};\tfrac{-{{x}^{2}}}{\nu } \right)=\frac{1}{2\sqrt{\tfrac{{{x}^{2}}}
{\nu }}}\left\{ {{\left( \frac{{{x}^{2}}}{v} \right)}^{\tfrac{1}{2}}}{\exp(\tfrac{-{{x}^{2}}}
{\nu })}h_{\tfrac{-1}{2}}^{\tfrac{-{{x}^{2}}}{\nu }}+ \right. \\
\left( \frac{v-1}{2} \right)\left[ {{\left( \frac{{{x}^{2}}}{v} \right)}^{\tfrac{1}{2}}}{\exp(\tfrac{-{{x}^{2}}}{\nu })}
h_{\tfrac{-1}{2}}^{\tfrac{-{{x}^{2}}}{\nu }}-\frac{{{\left( \frac{{{x}^{2}}}{v} \right)}^{\tfrac{1}{2}}}}{0!\frac{1}{2}} \right]+\\
\sum\limits_{i=1}^{\infty } \left\{ {{{\left( \frac{v-1}{2}+i \right)}^{2}}}\cdot \prod\limits_{j=1}^{i-1}
{\left( \frac{v-1}{2}+j \right)} \right. \\ \left.
\left[ {{\left( \frac{{{x}^{2}}}{v} \right)}^{\tfrac{1}{2}}}{\exp(\tfrac{-{{x}^{2}}}{\nu })}
h_{\tfrac{-1}{2}}^{\tfrac{-{{x}^{2}}}{\nu}}-\sum\limits_{k=0}^{i}{{{\left( -1 \right)}^{k}}
\frac{{{\left( \frac{{{x}^{2}}}{v} \right)}^{\tfrac{1}{2}+k}}}{k!\left( \frac{1}{2}+k \right)}} \right] \right\}. \tag{4.11}
\end{multline*}
[Proof for (4.11)]
This result demonstrates that the cumulative distribution of Student's t distribution can be expressed as an "$h$" function.
Welcome all math lovers!
