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!