Pade approximation to exp(x)

Pade approximations are rational polynomial approximations to $e^x$, i.e. of the form

$$e^x = \frac{p(x)}{q(x)} + O(x^m)$$

where $p(x)$ and $q(x)$ are polynomials in $x$.

Theorem: Given any integers $\alpha, \beta \ge 0$, there exists a UNIQUE function

$$\gamma_{\alpha\beta} = \frac{p_{\alpha\beta}}{q_{\alpha\beta}}, \quad q_{\alpha\beta}(0) = 1$$

which approximates $e^x$ to order $\alpha + \beta$. The explicit forms of $p_{\alpha\beta}$, $q_{\alpha\beta}$ are

$$\begin{eqnarray*} p_{\alpha\beta}(x) &=& \sum^\alpha_{k=0} {\alpha \choose k} \f... ...ha + \beta)!} x^k \\ q_{\alpha\beta}(x) &=& p_{\beta\alpha}(-x) \end{eqnarray*}$$

Moreover, $\gamma_{\alpha\beta}$ is (upto a rescaling of numerator and denominator by a non-zero multipication constant) the only member of $P_{\alpha\beta}$ of order $\alpha + \beta$ and no other function in $P_{\alpha\beta}$ may exceed this order. Here, $P_{\alpha\beta}$ is the set of all functions of the form $p_{\alpha\beta}/q_{\alpha\beta}$. Note that if $\beta=0$ then we recover the usual series expansion of $e^x$.

Some examples are

$$\begin{eqnarray*} \gamma_{10} &=& 1 + x \\ \gamma_{11} &=& \frac{ 1 + \frac{1}{... ...=& \frac{ 1 + \frac{1}{3}x}{ 1 - \frac{2}{3}x + \frac{1}{6} x^2} \end{eqnarray*}$$