Best estimates of the generalized Stirling formula

Applied Mathematics and Computation, 2010

The aim of this paper is to improve Ramanujan's formula for approximation of the factorial function, starting from Burnside's formula in contradistinction with the classical formula that starts from Stirling's formula.

Irish Mathematical Society Bulletin, 2016

A self-contained account of Stirling's formula for n! is presented that is based on definitions of the constants e and π that appear in it, and uses only the rudiments of the analysis of the convergence of numerical sequences, infinite series, and infinite products.

The American Mathematical Monthly, 1990

2014

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Some asymptotic expansions for the generalized Stirling functions and generalized Stirling numbers are established. In addition, four algorithms for calculating the Stirling numbers based on our generalization are also given. Tian-Xiao He Dept. Math, Illinois Wesleyan University Bloomington, IL, USA Asymptotic Expansions and Com...

MAT-KOL (banja Luka), 2018

In this paper are given the proof of important Stirling's formula and several hers interesting applications.

Archiv Der Mathematik, 2009

We prove in this paper that for every x ≥ 0, $$\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right) ^{x+\frac {1}{2}} < \Gamma(x+1)\leq\alpha\cdot\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right)^{x+\frac{1}{2}}$$ where ${\omega=(3-\sqrt{3})/6}$ and α = 1.072042464..., then $$\beta\cdot\sqrt{2\pi e}\cdot e^{-\zeta}\left(\frac{x+\zeta}{e}\right)^{x+\frac{1}{2}}\leq\Gamma(x+1) < \sqrt{2\pi e}\cdot e^{-\zeta}\left( \frac{x+\zeta}{e}\right)^{x+\frac{1}{2}},$$ where ${\zeta=(3+\sqrt{3})/6}$ 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.

The journal of combinatorial mathematics and combinatorial computing, 2013

Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Three algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.

arXiv (Cornell University), 2011

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Three algorithms for calculating the Stirling numbers based on our generalization are also given, which include a comprehensive algorithm using the characterization of Riordan arrays.

Computers & Mathematics With Applications, 2010

We introduce the Stirling's formula in a more general class of approximation formulas to extend the integral representation of Z. Liu [Tamsui Oxf. J. Math. Sci. 4 (2007) 389-392]. Finally, an accurate approximation for the factorial function is established. MSC 2000: 05A10; 41A60; 41A80; 26D15; 40A05

Asymptotic formulas for the generalized Stirling numbers of the second kind with integer and real parameters are obtained and ranges of validity of the formulas are established. The generalizations of Stirling numbers considered here are generalizations along the line of Hsu and Shuie's unified generalization.