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Abstract: Using a functional substitution and Faà di Bruno formula, Stirling asymptotic series approximation to the Γ(z) function is converted into a new one with better convergence properties. The new formula is compared with those of Stirling, Laplace and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of 'correction' terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling's (maximum relative error smaller than 1e-10 for real arguments greater or equal to 10).
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Full text (PDF, 144 kBytes).
Note added on 28 Dec 2008: This work has been now complemented by remainder estimates for the asymptotic series.
Click here for the updated version.
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Editor's Note:
Gergõ Nemes is a young Hungarian mathematician who is just starting in earnest his formal mathematical education. He has tried a functional transformation of the Stirling's asymptotic series going, without getting lost, through the messy chore of Faà di Bruno formula and came up with an interesting novel series. While Stirling's series is asymptotic (i.e., not convergent), the Nemes expansion might be actually convergent (at least for arguments greater than 1). Though this is not yet quite clear, I find very interesting the implication that a functional transformation might convert a non-convergent asymptotic series into a convergent one.
I also like the 'closed-form' formula in Section 5, which Gergõ just happened to notice and pick up while comparing his formula with that of Laplace. Considering its simplicity, it is amazingly accurate.
If you wish, download the Matlab program used to numerically test the various approximations and draw Figure 1.
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