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How we can express this series $F(z)=\sum_{n=0}^\infty \frac{z^n}{(a)_nn!}$ in terms of Gauss' hypergeometric function?

where $(a)_n$ denotes the Pochhammer symbol.

Thanks in advance

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    summation from zero to infinity2012-05-15

1 Answers 1

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The above series represents generalized hypergeometric function, but it is not related to the Gauss's hypergeometric function.

This particular series represents ${}_0F_{1}(;a;z)$ (see here), and is related to Bessel function: $ \sum_{n=0}^\infty \frac{z^n}{n!} \frac{1}{(a)_n} = \sum_{n=0}^\infty \frac{z^n}{n!} \frac{\Gamma(a)}{\Gamma(a+n)} = \Gamma(a) z^{\frac{1-a}{2}} I_{a-1}\left(2 \sqrt{z} \right) $ where $I_\nu(z)$ denotes the modified Bessel function of the first kind.

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    @MAK Since you are new to math.SE, and if you like this site, please browse through the [FAQ](http://math.stackexchange.com/faq). Great/interesting questions and answers get recognized by up-voting (clicking the up-arrow to the left of the question/answer). One of the answers extended ultimately becomes accepted (by clicking the tick symbol to the left of the answer), if it indeed answered the question. Accepting answers [is important](http://meta.math.stackexchange.com/questions/3399/why-should-we-accept-answers) part of site workflow.2012-05-16