Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration

Abstract

We give an apparently new proof of Stirling’s original asymptotic formula for the behavior of $\text{ln}\hspace{0.17em}z!$ for large z. Stirling’s original formula is not the formula widely known as “Stirling’s formula”, which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\text{ln}\hspace{0.17em}z!$ over $\mathbb{C},$ when coupled with the sequence acceleration method known as Levin’s u-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.

Keywords:

• Asymptotic series
• sequence acceleration
• Levin’s u-transform
• Stirling’s formula

Notes

1 Of course, there is no hope of changing the popular meaning of the name “Stirling’s formula”.

2 https://isc.carma.newcastle.edu.au. Remark: The ISC is currently down because a security flaw was found. Discussion is under way as to how or if this can be resolved.

3 Correctly, in the sense of Euler summation, taking $1+r+r^2+\cdots =1/(1-r)$ even if $\left|r\right|>1$ by redefining what the infinite sum actually means: see e.g. [Hardy 00], for more classical work on making sense of divergent series.

4 Except of course for rounding error. We do not attempt a numerical analysis here, which appears involved. The main difficulty is predicting the number of arithmetic operations.