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Windschitl type approximation formulas for the gamma function.

Zhen-Hang Yang^1,2, Jing-Feng Tian¹

¹1College of Science and Technology, North China Electric Power University, Baoding, P.R. China.

Journal of Inequalities and Applications

|October 27, 2018

Summary

This summary is machine-generated.

Researchers developed four new Windschitl-type approximation formulas for the gamma function. These formulas demonstrate useful properties, leading to new inequalities for gamma and factorial functions.

Keywords:

Convexity Gamma function Inequality Monotonicity Windschitl type approximation formula

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Area of Science:

Mathematical Analysis
Number Theory

Background:

The gamma function is a fundamental concept in mathematical analysis, extending the factorial function to complex and real numbers.
Approximation formulas are crucial for efficiently computing and analyzing the behavior of the gamma function.

Purpose of the Study:

To introduce four novel Windschitl-type approximation formulas for the gamma function.
To investigate the properties (monotonicity and convexity) of functions involving these new approximations.
To derive new inequalities and refine existing ones for the gamma and factorial functions.

Main Methods:

Development of new approximation formulas based on Windschitl's work.
Analytical techniques to prove monotonicity and convexity properties.
Derivation of inequalities using the established properties.

Main Results:

Four new Windschitl-type approximation formulas for the gamma function were successfully derived.
The functions incorporating these formulas exhibit desirable properties like monotonicity and convexity.
New inequalities for the gamma and factorial functions were established.

Conclusions:

The novel approximation formulas offer valuable tools for gamma function analysis.
The proven properties and derived inequalities contribute to a deeper understanding of the gamma and factorial functions.
This work provides a new perspective and strengthens existing results in the field.