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Made the series representation more explicit by way of the gamma function. This way future readers don't have to guess what the exact representation of the generalized hypergeometric function looks like. |
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Line 39:
:<math>\begin{align}
(a)_0 &= 1, \\
(a)_n &= a(a+1)(a+2) \cdots (a+n-1) = \frac{\Gamma(a+n)}{\Gamma(a)}, && n \geq 1,
\end{align}</math>
where <math>\Gamma(x)</math> represents the [[Gamma function|gamma function]], this can be written
:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!} = \frac{\Gamma(b_1)\cdots \Gamma(b_q)}{\Gamma(a_1)\cdots \Gamma(a_p)} \sum_{n=0}^{\infty} \frac{\Gamma(n+a_1)\cdots \Gamma(n+a_p)}{\Gamma(n+b_1)\cdots \Gamma(n+b_q)} \frac{z^n}{n!}.</math>
(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)
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