BEST ESTIMATES OF THE GENERALIZED STIRLING FORMULA CRISTINEL MORTICI Abstract. The aim of this paper is to introduce an approxima- tions family of the factorial function that contains Stirlings formula, Burnsides formula and Gospers formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas. MSC: 30B10, 41A58, 33B15 Keywords: approximations; speed of convergence; factorial function; gamma function. 1. Introduction Undoubtedly the most known and most used formula for approximation of the large factorials is the following formula n!  p 2n  n e  n = p 2e  n e  n+1=2 =  n ; now known as Stirlings formula. It is named after the English mathemati- cian James Stirling (1692-1770), who treated it in his famous work Methodus Di/erentialis, published in 1730, e.g., [17]. If in probabilities or statistical physics, such approximations are reasonable, in pure mathematics, better formulas are required. It is true that Stirlings formula can be indenitely improved, but a sacrice of simplicity, and this is why Stirlings formula survived so long. Other recently approximation formulas for the factorial function, or for Eulers gamma function were stated, e.g., in [2-5, 7-14]. A slightly more better result is the following n!  p 2  n +1=2 e  n+1=2 =  n , due to Burnside [1]. Motivated by these approximations  n and  n ; Mortici [8] introduced the following approximations family n!  p 2e  e p  n + p e  n+ 1 2 ; (1) 1