Abstract

The goal of this paper is to prove the identity where and where is the Gamma function defined by and is the Euler-Mascheroni constant.

1. Introduction

The Euler-Gamma function is defined by, valid in the entire complex plane, except at where it has simple poles ¹. It can also be seen as a generalization of the factorial on the positive integers to the rationals. Indeed the Gamma function (See ^{1, 2}) satisfies the functional equation and

so that in the case and is a positive integer, then we have the expression The Gamma function still remains valid for arguments in the range by the equation

It also has the canonical product representation (See ³)

valid for . The gamma function also has very key properties, most notably the duplication and the complementary property (reflexive formula), which are given respectively as

and

For many more of these properties, the reader is encouraged to see ¹. The Gamma function is also inextricably linked to some very interesting functions. Consider the digamma function ¹, the logarithmic derivative of the Gamma function defined by

The Gamma funtion has spawn a great deal of research and out of which has led to the discovery of many beautiful identities and inequalities. More recently the gamma function has been studied by Alzer and many other authors. For more results on the gamma function, see ^{2, 3}. In this paper, however, we prove a certain identity related to the Gamma function.

2. Main Theorem

Theorem 2.1. For any , we have

where

and

where is the Gamma function defined by and

is the Euler-Mascheroni constant.

Proof. Let be a real-valued function, contineously differentiable on the interval and for all . Then we set

for . In the simplest case, we choose , since it satisfies the hypothesis. Thus . By application of integration by parts, we find that

where and are convergent. More precisely, we can write in a closed form as

Now, since is analytic in the half plane , it follows by the convergence of that

On the other hand . Arranging terms and comparing both results we find that

(2.1)

Using the following identities involving the Gamma function ¹

(2.2)

(2.3)

the remaining task is to arrange the terms and apply these identities and identify the function and . We leave the remaining task to the reader to verify.

Remark 2.2. Now we examine some immediate conequences of the above result, in the following sequel.

Corollary 1. The identity

where

and

remains valid.

Proof. Let us set in Theorem 2.1. Then it follows that

where we have used the relation ¹. The proof is completed by computing and given in Theorem 2.1.

Corollary 2. The identity

is valid, where

and

Proof. The result follows by setting in Theorem 2.1, and computing and .

Corollary 3. For any integer , the inequality

where

and

is valid.

Proof. If is an integer, then Theorem 2.1 reduces to

and the result follows immediately by applying the triangle inequality.

3. Final Remarks

In this paper we proved an identity related the reciprocal of the gamma function. Consequently, we obtained the following identities

where

and

and

with

and

Conict of Interest

The author declares that he has no conict of interest in relation to this article.

Ethical Approval

This article does not contain any studies with human participants or animals per-formed by the author.

References