scholarly journals A generalization of Mortici lemma

2012 ◽  
Vol 21 (2) ◽  
pp. 129-134
Author(s):  
VASILE BERINDE ◽  
Keyword(s):  
Gamma Function ◽  
Asymptotic Expansions ◽  
Asymptotic Integration ◽  
Stirling Formula ◽  
Monotonic Functions ◽  
Factorial Function ◽  
Approximation Formulas ◽  
Appl Math ◽  
Best Estimates

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.

scholarly journals Improvements of asymptotic approximation formulas for the factorial function

2019 ◽  
Vol 13 (3) ◽  
pp. 895-904
Author(s):  
Tomislav Buric
Keyword(s):  
Gamma Function ◽  
Asymptotic Expansions ◽  
Asymptotic Approximation ◽  
Factorial Function ◽  
Approximation Formulas

Asymptotic expansions of the gamma function are studied and new accurate approximations for the factorial function are given.

INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 453-458
Author(s):  
JENICA CRINGANU
Keyword(s):  
Gamma Function ◽  
Monotonic Function ◽  
Completely Monotonic Function ◽  
Gamma Functions ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
Appl Math

We use properties of the gamma function to estimate the products$\prod _{k=1}^{n}(4k-3)/4k$and$\prod _{k=1}^{n}(4k-1)/4k$, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’,Tamkang J. Math.36(4) (2005), 303–307] and Morticiet al.[‘Completely monotonic functions and inequalities associated to some ratio of gamma function’,Appl. Math. Comput.240(2014), 168–174].

scholarly journals Two asymptotic expansions for gamma function developed by Windschitl’s formula

2018 ◽  
Vol 16 (1) ◽  
pp. 1048-1060 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Gamma Function ◽  
Asymptotic Expansions ◽  
Bernoulli Number ◽  
Closed Form Expression ◽  
Approximation Formula ◽  
Form Expression ◽  
Approximation Formulas ◽  
Windschitl’S Formula

AbstractIn this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have$$\begin{array}{} \displaystyle {\it \Gamma} \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1} {\frac{a_{n}}{x^{2n-1}}}+R_{n}\left( x\right) \right) \end{array}$$with$$\begin{array}{} \displaystyle \left\vert R_{n}\left( x\right) \right\vert \leq \frac{\left\vert B_{2n}\right\vert }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{array}$$for all x > 0, where an has a closed-form expression, B2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.

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