Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.