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Do we have a general solution infinite series for the gamma function? 
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#1
Feb2414, 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, $${\Gamma(k)}=\sum_{n=0}^{\infty} f(n,k)$$? 


#2
Feb2414, 03:53 PM

Sci Advisor
P: 6,039



#3
Feb2414, 04:08 PM

P: 511

Okay I see the following two general solutions, $$\Gamma(t)=x^t \sum_{n=0}^{\infty} \frac{L_n^{(t)}(x)}{t+n}$$ $$\Gamma(z) = 1+\sum_{k=1}^\infty\frac{\Gamma^{(k)}(1)}{k!}(z1)^{k}$$ 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 (z1)<1 so it is pretty limited. Are there any others? 


#4
Feb2414, 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.



#5
Feb2414, 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.



#6
Feb2414, 09:08 PM

P: 511




#7
Feb2514, 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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