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

Question

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

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.