Computation of the gamma, digamma, and trigamma functions
SIAM Journal on Numerical Analysis
On the asymptotic representation of the Euler gamma function by Ramanujan
Journal of Computational and Applied Mathematics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A class of integral approximations for the factorial function
Computers & Mathematics with Applications
The asymptotic series of the generalized Stirling formula
Computers & Mathematics with Applications
A new Stirling series as continued fraction
Numerical Algorithms
Improved asymptotic formulas for the gamma function
Computers & Mathematics with Applications
New approximation formulas for evaluating the ratio of gamma functions
Mathematical and Computer Modelling: An International Journal
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In this paper, we establish two families of approximations for the gamma function:$$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$, $b$, $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes' formulae, which, to our knowledge, are better than other approximations in the literature.