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Examples

Many well-known series, including those that are encountered in
elementary calculus courses, are hypergeometric. For instance,
McLauren series for $e^x$:

Example 1  

$$e^x = 1 + x + \frac{x^2}{2!} + \cdots = \sum_{k=0}^\infty \frac{x^k}{k!}$$

The ratio $F(x,k+1)/F(x,k) = \frac{x}{k+1}$, is clearly a rational
function of $x$ and $k$, so we can write:

$$e^x = {}_{0}\mathrm{F}_{0} \left[ \genfrac{}{}{0pt}{}{-}{-} ; {x} \right]$$

Here are two more examples:

Example 2  

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{k=0}^\infty \... ...)!} = x\sum_{k=0}^\infty \frac{(-1)^k x^{2k} }{(2k+1)!} \frac{F(n,k+1)}{F(n,k)}$$

$$= \frac{-x^2}{(2k+2)(2k+3)} = \frac{-x^2}{4} \frac{1}{(k+1)(k+3/2)},$$

so

$$\sin x = x {}_{0}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{-}{3/2} ; {\frac{-x^2}{4}} \right] .$$

Example 3  

$$\arctan x = x- \frac{x^3}{3} + \frac{x^5}{5}-\dots = \sum_{k=0}^\infty \frac{... ...}{2k+1} = x \sum_{k=0}^\infty \frac{(-1)^kx^(2k)}{2k+1} \frac{F(n,k+1)}{F(n,k)}$$

$$= \frac{-x^2 (2k+1)}{2k+3} \frac{k+1}{k+1},$$

so

$$\arctan x = x {}_{2}\mathrm{F}_{1} \left[ \genfrac{}{}{0pt}{}{1/2,1}{3/2} ; {-x^2} \right] .$$