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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.

Hedonist
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  • 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 – njuffa Feb 16 '15 at 20:54
  • Gordon, 1994, A Stochastic Approach to the Gamma Function, in Theorem 5 generalizes the easy to use bounds for the factorial from Robbins (1955) to the entire gamma function. It gives precise upper and lower bounds for the gamma function in terms of Stirling's approximation. – hasManyStupidQuestions May 08 '21 at 03:51
  • By $(5.6.8)$, $x^a \le \frac{{\Gamma (x + a)}}{{\Gamma (x)}}$ for $x>0$ and $a\ge 1$. – Gary Mar 21 '24 at 02:56
  • See also https://math.stackexchange.com/q/3076639 – Gary Mar 22 '24 at 04:45

1 Answers1

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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 $n^\alpha$.

rubik
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  • 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. – Hedonist Feb 15 '15 at 18:47
  • @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. – rubik Feb 15 '15 at 18:57