Formulas for factorial 𝑁

Murray, Francis J.

doi:10.1090/S0025-5718-1982-0669657-1

Formulas for factorial

Author: Francis J. Murray
Journal: Math. Comp. 39 (1982), 655-662
MSC: Primary 33A15; Secondary 39A70, 41A60
DOI: https://doi.org/10.1090/S0025-5718-1982-0669657-1
MathSciNet review: 669657
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Abstract: Burnside's and Stirling's formulas for factorial N are special cases of a family of formulas with corresponding asymptotic series given by E. W. Barnes in 1899. An operational procedure for obtaining these formulas and series is presented which yields both convergent and divergent series and error estimates in the latter case. Two formulas of this family have superior accuracy and the geometric mean is better than either.

[1] W. Burnside, "A rapidly convergent series for $\operatorname{Log}\;N$ !," Messenger Math., v. 46, 1917, pp. 157-159.
[2] Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
[3] Philip Franklin, A Treatise on Advanced Calculus, John Wiley & Sons, Inc., New York, 1940. MR 0002571
[4] M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge University Press, New York, 1960. MR 0115085
[5] Kenneth S. Miller, An introduction to the calculus of finite differences and difference equations, Dover Publications, Inc., New York, 1966. MR 0206540
[6] J. R. Wilton, "A proof of Burnside's formula for ${\operatorname{Log}}\;\Gamma (x + 1)$ and certain allied properties of the Riemann $\zeta$ -function," Messenger Math., v. 52, 1922, pp. 90-95.