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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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What have you attempted? What are the summation bounds? –  Sasha May 15 '12 at 21:32
    
summation from zero to infinity –  MAK May 15 '12 at 21:38
    
Why did you remove the homework tag? Is this not a homework question? –  Sasha May 16 '12 at 13:03
    
Actually, it was not a homework. Sorry –  MAK May 16 '12 at 13:24

1 Answer 1

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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Thank you very mauch –  MAK May 16 '12 at 6:47
    
@MAK Since you are new to math.SE, and if you like this site, please browse through the 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 part of site workflow. –  Sasha May 16 '12 at 13:14

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