# Is this series expressible in terms of Gauss' hypergeometric function?

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.

-
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

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.