# How to find: $f^{\alpha}_n(x)=1+\sum_{k=1}^{\infty} \frac{\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$

I am looking for a closed-form solution to $$f^{\alpha}_n(x)=a_0+\sum_{k=1}^{\infty} \frac{a_k\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$$ where we can take $$a_k = 1, \forall k\in [0,\infty)$$. Maybe a definite integral representation or something simpler than the initial computational complexity of this?

• Hi and welcome to the Math.SE. Could you provide a little bit of context? Your question looks interesting per se, but if you add some motivation, it will attract the attention of many more members. – Daniele Tampieri Apr 19 at 12:44
• This is just the ordinary hypergeometric function (which is an infinite summation). – Claude Leibovici Apr 19 at 13:22