This is probably an easy/trivial question, but being an engineer my background on maths isn't very strong and I'm unable to find the solution to this problem. I hope anyone can help.
I have been able to approximate a certain function $f(x)$ by a power series as follows:
$$f(x) = \sum_{n=0}^\infty \frac{\Gamma(1-a)}{\Gamma(1+n) \Gamma(1-a-n)} \ x^n$$
where $\Gamma(x)$ is the gamma function and $a$ is a positive real number.
The domain of the function $f(x)$ is the interval $[0, \infty)$ but I can only use the power series for values of the argument $x < 1$. From what I have found after some research, it should be possible to calculate an analytical continuation (that is, obtain another expression from this one) that would be valid for $x>1$.
I have tried to understand how this is done but I have been unable to understand how it works. I guess it must be a basic procedure but unfortunately I have been unable to succeed. Any help would be much appreciated.