# Do we have a general solution infinite series for the gamma function?

1. ### mesa

593
Does anyone know if we currently have an infinite series summation general solution for the gamma function such as,

$$\frac{1}{\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$

or,

$${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$?

6,569
3. ### mesa

593
How I miss these things in wikipedia I will never know...
Okay I see the following two general solutions,

$$\Gamma(t)=x^t \sum_{n=0}^{\infty} \frac{L_n^{(t)}(x)}{t+n}$$

$$\Gamma(z) = 1+\sum_{k=1}^\infty\frac{\Gamma^{(k)}(1)}{k!}(z-1)^{k}$$

For the first one I am unfamiliar with Laguerre polynomials. Does this cause any restrictions on the input of the function?

The second one only works for the absolute value of (z-1)<1 so it is pretty limited.

Are there any others?

4. ### mesa

593
Well I have something much better, if anyone knows of anything else please respond.

5. ### eigenperson

160
$1/\Gamma$ is an entire function, so its Taylor series converges to it everywhere. The coefficients are given at Wikipedia's article on the reciprocal gamma function.

6. ### mesa

593
Are you referring to the Taylor series as another general solution except in the form of $1/\Gamma$?

7. ### mesa

593
Update, my function now also works for imaginary values in addition to the real number line.