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

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mesa
#1
Feb24-14, 01:40 PM
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,

$${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$?
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mathman
#2
Feb24-14, 03:53 PM
Sci Advisor
P: 6,039
http://en.wikipedia.org/wiki/Gamma_function

Try the above.
mesa
#3
Feb24-14, 04:08 PM
P: 511
Quote Quote by mathman View Post
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?

mesa
#4
Feb24-14, 08:55 PM
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.
eigenperson
#5
Feb24-14, 09:00 PM
P: 160
[itex]1/\Gamma[/itex] 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.
mesa
#6
Feb24-14, 09:08 PM
P: 511
Quote Quote by eigenperson View Post
[itex]1/\Gamma[/itex] 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 [itex]1/\Gamma[/itex]?
mesa
#7
Feb25-14, 12:53 PM
P: 511
Update, my function now also works for imaginary values in addition to the real number line.


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