\begin{definition}[Definition:Hypergeometric Series]
A '''hypergeometric series''' is a power series:
:$\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{n \mathop \ge 0} \beta_n z^n$
in which the ratio of successive coefficients is a rational function of $n$:
:$\dfrac {\beta_{n + 1} } {\beta_n} = \dfrac {\map A n} {\map B n}$
where $\map A n$ and $\map B n$ are polynomials in $n$.
{{expand|Needs to include a derivation of the notation ${}_p F_q \tuple {a_1, \ldots, a_p; b_1, \ldots, b_q; z}$}}
{{questionable|{{BookLink|The Penguin Dictionary of Mathematics|David Nelson|ed = 4th|edpage = Fourth Edition}} defines this where the coefficient $\beta_n$ is specifically $\dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!}$}}
\end{definition}