Arch. Math. 93 (2009), 37–45 c  2009 Birkh¨auser Verlag Basel/Switzerland 0003-889X/09/010037-9 published online July 7, 2009 DOI 10.1007/s00013-009-0008-5 Archiv der Mathematik An ultimate extremely accurate formula for approximation of the factorial function Cristinel Mortici Abstract. We prove in this paper that for every x ≥ 0, √ 2πe · e -ω  x + ω e  x+ 1 2 < Γ(x + 1) ≤ α · √ 2πe · e -ω  x + ω e  x+ 1 2 where ω = (3 - √ 3)/6 and α =1.072042464..., then β · √ 2πe · e -ζ  x + ζ e  x+ 1 2 ≤ Γ(x + 1) < √ 2πe · e -ζ  x + ζ e  x+ 1 2 , where ζ = (3 + √ 3)/6 and β =0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form. Mathematics Subject Classification (2000). Primary: 40A25; Secondary 26D07. Keywords. Factorial function, Gamma function, Digamma function, Numeric series, Stirling’s formula, Burnside’s formula and inequalities. 1. Introduction. Stirling’s formula and its generalizations have a large class of applications in science as in statistical physics or probability theory. In con- sequence, it has been deeply studied by a large number of authors, due to its practical importance. Stirling’s formula: n! ≈  n e  n √ 2πn = σ n , (1.1) is an approximation for big factorials. In fact, the formula (1.1) was discovered by the French mathematician Abraham de Moivre (1667–1754) in the form n! ≈ constant ·  n e  n √ n and the Scottish mathematician James Stirling (1692–1770) discovered the constant √ 2π in the previous formula. For proofs and other details see [6].