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

 P: 511 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, $${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$?
 Sci Advisor P: 6,039 http://en.wikipedia.org/wiki/Gamma_function Try the above.
P: 511
 Quote by mathman http://en.wikipedia.org/wiki/Gamma_function Try the above.
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?

 P: 511 Do we have a general solution infinite series for the gamma function? Well I have something much better, if anyone knows of anything else please respond.
 P: 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.
P: 511
 Quote by eigenperson $1/\Gamma$ 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.
Are you referring to the Taylor series as another general solution except in the form of $1/\Gamma$?
 P: 511 Update, my function now also works for imaginary values in addition to the real number line.

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