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I am interested in strong upper and lower bounds on $\frac{\Gamma(n+\alpha)}{\Gamma(n)},$ where $n$ is a large non-integral number and $\alpha$ is a small constant like $3.5.$ I know the answer is approximately $n^\alpha$ but I want multiplicative guarantees on how good this approximation is, both upper and lower bounds. I suppose there is a version of Stirling's formula that can give me what I want.

Thanks.

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  • $\begingroup$ Have you tried a thorough literature search? There are many papers that deal with gamma function inequalities. The closest I could find to your question (unfortunately not providing an answer for the range of $a$ you desire as it restricts itself to $\frac {\Gamma (x+1)}{\Gamma (x+s)}$, $0 \lt s \lt 1$ is: D. Kershaw, Some Extensions of W. Gautschi's Inequalities for the Gamma Function, Math. Comput., vol. 41, no. 164, Oct. 1983, pp. 607-611 $\endgroup$ – njuffa Feb 16 '15 at 20:54
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For any complex $z$, we have that $$\Gamma(z) = \sqrt{\frac{2\pi}z}\bigg(\frac z e\bigg)^z\left(1 + \mathcal O\left(\frac1z\right)\right).$$ Since you said $n$ is large, we can take $$\Gamma(n) \approx \sqrt{\frac{2\pi}n}\bigg(\frac n e\bigg)^n.$$ Applying this to your function, we get, after quite a few basic algebraic manipulations, $$\frac{\Gamma(n + \alpha)}{\Gamma(n)} \approx \bigg(1 + \frac\alpha n\bigg)^{n - \frac12}\left(\frac{n + \alpha}e\right)^\alpha\tag1$$

By taking as lower and upper bounds $$\begin{align}\mathcal L(n, \alpha) &= \bigg(1 + \frac\alpha n\bigg)^{n - 1}\left(\frac{n + \alpha}e\right)^\alpha\\ \mathcal U(n, \alpha) &= \bigg(1 + \frac\alpha n\bigg)^n\left(\frac{n + \alpha}e\right)^\alpha \end{align}$$ you obtain really strong bounds. Approximation $(1)$ is much better than $n^\alpha$.

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  • $\begingroup$ Thanks. Can you tell me the precise upper and lower bounds on $\Gamma(n)$ that you used to obtain your upper and lower bounds? What I mean is that the Stirling formula with the $O$ notation is not sufficient for my purposes since I need a non-asymptotic handle on the term. $\endgroup$ – Hedonist Feb 15 '15 at 18:47
  • $\begingroup$ @Hedonist Ahh I'm sorry but I cannot help you here. After I derived $(1)$ I went on with sort of "empirical trials", since you didn't give precise conditions on $n$ and $\alpha$. I did a few calculations and the bounds seemed to hold for a variety of cases. I also tried to hack together a proof to show the the bounds actually held, but it wasn't complete nor usable. $\endgroup$ – rubik Feb 15 '15 at 18:57

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