Approximation for the gamma function via the tri-gamma function

Xu You,Xiaocui Li

doi:10.1186/s13660-017-1585-7

Abstract

In this paper, we present a new sharp approximation for the gamma function via the tri-gamma function. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.

Highlights

1 Introduction It is well known that we often need to deal with the problem of approximating the factorial function n! and its extension to real numbers called the gamma function, defined by
The asymptotic expansion of (x + 1) via the tri-gamma function can be generalized to the general cases by the arguments in [2] as follows
Using Theorem 1, we provide some inequalities for the gamma function

Summary

Introduction

Its extension to real numbers called the gamma function, defined by (x) = tx–1e–t dt, Re(x) > 0, and the logarithmic derivatives of (x) are called the psi-gamma functions, denoted by d (x) ψ(x) = ln (x) = . For x > 0, the derivatives ψ (x) are called the tri-gamma functions, while the derivatives ψ(k)(x), k = 1, 2, 3, . The asymptotic expansion of (x + 1) via the tri-gamma function can be generalized to the general cases by the arguments in [2] as follows.

Results

Conclusion