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My friend came to see me regarding some calculation in probability where he would like to know if it is possible to solve for a variable analytically under the gamma function. By this I mean say we are given the value of some quantity $x$, such that

$x = \dfrac{\Gamma(1 + \frac{2}{k})}{\left(\Gamma(1 + \frac{1}{k})\right)^2}$

I would like to solve this for some real number $k$. I can use the factorial to manipulate this expression and get

$x = \dfrac{(\frac{2}{k})!}{{\left(\left(\frac{1}{k}\right)!\right)}^2 }$

If (a big if) I can make the substitution $n = \frac{1}{k}$, this would be equal to the central binomial coefficient. However I don't think this is possible as $\frac{1}{k}$ is not an integer in general.

What else can I do? Thanks.

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@DJC Why is there a $2 + 2/k$ in there? Did you mean $1 + 2/k$? –  fpqc Sep 28 '11 at 3:48
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I will delete my other comment. Less explicitly, but helpful (I hope): The Beta-function is related to the Gamma function by $$B(x_1,x_2) = \frac{\Gamma(x_1) \Gamma(x_2)}{\Gamma(x_1 + x_2)}.$$ Setting $x_1 = x_2 = 1 + \frac{1}{k}$ gives $$\frac{1}{x} = \left(2 + \frac{2}{k}\right) B\left(1 + \frac{1}{k}, 1 +\frac{1}{k}\right).$$ –  JavaMan Sep 28 '11 at 3:59
    
Thanks, I wills see what I can work with from there. –  fpqc Sep 28 '11 at 4:46
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Considering how difficult it is to solve similar equations for Factorials, solving these gamma function equations are certainly not possible analytically. For most purposes using Stirling's approximation to solve for $x$ numerically should suffice. –  Ragib Zaman Sep 28 '11 at 7:55
    
A closed-form solution looks unlikely to me. You'll definitely have to use a numerical method like Newton-Raphson for this. –  J. M. Sep 28 '11 at 9:29
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1 Answer

up vote 3 down vote accepted

Let $u = \frac{1}{k}$. Using duplication formula for the numerator: $$ \Gamma\left( 2 u + 1\right) = \frac{4^u}{\sqrt{\pi}} \Gamma\left(u + 1\right)\Gamma\left(u + \frac{1}{2}\right) $$ we rewrite your equation as $$ x = \frac{4^u}{\sqrt{\pi}} \frac{\Gamma\left(u + \frac{1}{2}\right)}{\Gamma\left(u + 1\right)} $$ This form is helpful if $u$ is large, because $\frac{\Gamma\left(u + \frac{1}{2}\right)}{\Gamma\left(u + 1\right)} \sim \sqrt{u} + o(1)$

Thus if $x$ is large, equation becomes $x = 4^u \sqrt{\frac{u}{\pi}}$, which can be solved exactly $u = \frac{W(4 x^2 \pi \log x )}{4 \log 2}$, where $W$ is the Lambert W function.

If $u$ is small, series expansion works well: $$ \frac{4^u}{\sqrt{\pi}} \frac{\Gamma\left(u + \frac{1}{2}\right)}{\Gamma\left(u + 1\right)} \sim 1 + \frac{\pi^2}{6} u^2 - 2 \zeta(3) u^3 + \frac{19 \pi^4}{360} u^4 + o(u^4) $$

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