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

  1. Feb 24, 2014 #1
    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. jcsd
  3. Feb 24, 2014 #2

    mathman

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  4. Feb 24, 2014 #3
    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?
     
  5. Feb 24, 2014 #4
    Well I have something much better, if anyone knows of anything else please respond.
     
  6. Feb 24, 2014 #5
    [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.
     
  7. Feb 24, 2014 #6
    Are you referring to the Taylor series as another general solution except in the form of [itex]1/\Gamma[/itex]?
     
  8. Feb 25, 2014 #7
    Update, my function now also works for imaginary values in addition to the real number line.
     
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