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

  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. mathman

    mathman 6,435
    Science Advisor
    Gold Member

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