Windschitl type approximation formulas for the gamma function

Zhen-Hang Yang,Jing-Feng Tian

doi:10.1186/s13660-018-1870-0

Zhen-Hang Yang, Jing-Feng Tian

Open Access

https://doi.org/10.1186/s13660-018-1870-0

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Abstract

In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.

Highlights

For x > 0, the classical Euler’s gamma function and psi function ψ are defined by (x) = ∞tx–1e–t dt and ψ(x) = (x), (x) (1.1)respectively
6 Conclusion In this paper, we provide four Windschitl type approximation formulas for the gamma function, and prove that those functions, involving the gamma function and Windschitl type functions, have good properties, including monotonicity and convexity
It is worth mentioning that our proofs of Theorems 1–5 are subtle and interesting, since the approximations deal with the gamma and hyperbolic sine functions, and it is difficult to establish their monotonicity and convexity by usual methods

Summary

Introduction

For x > 0, the classical Euler’s gamma function and psi (digamma) function ψ are defined by (x) = ∞. The gamma function has various important applications in many branches of science. For this reason, scholars strive to find various better approximations for the factorial or gamma function by using different ideas and techniques, for instance, Ramanujan [1, p. Some properties of the remainders of certain approximations for the gamma function can be found in [4, 16, 23, 30,31,32,33,34,35]. We are interested in Windschitl’s approximation formula (see [36]) given by

Objectives

Conclusion