Formulas for factorial 𝑁

Murray, Francis J.

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

Formulas for factorial $N$

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.

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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
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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.