超几何函数

\begin{equation} {_pF_q}(a_1,\dots, a_p; b_1, \dots, b_q; z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots (a_p)_n}{(b_1)_n\dots(b_q)_n} \frac{z^n}{n!}~. \end{equation}

\begin{equation} _1F_1(a; b; z) = \sum_{n=0}^\infty \frac{(a)_n}{(b)_n} \frac{z^n}{n!}~, \end{equation}

\begin{equation} \begin{aligned} {_1F_1}(a; b; z) &= \frac{(-1)^a\Gamma(b)}{\Gamma(b-a)} \sum_{n=0}^\infty (-1)^n\frac{(a)_n (a-b+1)_n}{n!} z^{-n-a}\\ &+ \frac{\Gamma(b)}{\Gamma(a)} \sum_{n=0}^\infty \frac{(b-a)_n (1-a)_n}{n!} z^{-n+a-b} \mathrm{e} ^z~, \end{aligned} \end{equation}

\begin{equation} {_1F_1}(a; b; z) = 1 + \frac{az/b}{1+\dots}\ \frac{-c_1 z}{1 + c_1 z + \dots}\ \frac{-c_2 z}{1 + c_2 z +\dots}~, \end{equation}

\begin{equation} c_n = \frac{a + n}{(n+1)(b + n)}~. \end{equation}